LMFDB - Newform orbit 304.8.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$304 = 2^{4} \cdot 19$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	304.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:	$94.9650477472$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 6373x^{4} - 12403x^{3} + 11165936x^{2} + 51537728x - 4683020288$ x^6 - x^5 - 6373x^4 - 12403x^3 + 11165936x^2 + 51537728x - 4683020288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{10}\cdot 3$
Twist minimal:	no (minimal twist has level 152)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 - 5) q^{3} + ( - \beta_{2} - \beta_1 - 7) q^{5} + (\beta_{5} - 4 \beta_1 + 152) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots - 38) q^{9} + ( - 3 \beta_{5} - \beta_{4} + \cdots - 652) q^{11}+ \cdots + (4717 \beta_{5} - 2113 \beta_{4} + \cdots + 493568) q^{99}+O(q^{100})$ q + (b1 - 5) * q^3 + (-b2 - b1 - 7) * q^5 + (b5 - 4b1 + 152) q^7 + (b5 - b4 + b3 + 2b2 - 5b1 - 38) * q^9 + (-3b5 - b4 + 3b3 - 6b2 - 19b1 - 652) * q^11 + (3b5 + 7b4 - 2b3 + 17b2 + 62b1 + 1063) q^13 + (-11b5 + 2b4 - 11b3 - 16b2 - 71b1 - 1394) q^15 + (-8b5 - 9b4 - 10b3 + 30b2 + 189b1 - 1185) q^17 + 6859 * q^19 + (-34b5 - 8b4 - 13b3 + 37b2 + 310b1 - 9131) q^21 + (-b5 + 14b4 - 16b3 + 45b2 - 419b1 + 7337) * q^23 + (-33b5 + 13b4 + 31b3 + 42b2 + 717b1 - 8117) q^25 + (-11b5 + 35b4 - 35b3 + 140b2 - 1449b1 - 1221) q^27 + (-28b5 + 47b4 + 37b3 + 27b2 + 335b1 - 22407) q^29 + (-13b5 - 27b4 + 137b3 + 282b2 - 1288b1 + 16150) q^31 + (5b5 + 181b4 - 97b3 - 2b2 - 530b1 - 35408) q^33 + (-29b5 - 98b4 - 21b3 + 344b2 - 1486b1 + 33930) q^35 + (173b5 - 486b4 + 125b3 - 12b2 - 433b1 - 9412) q^37 + (89b5 - 233b4 + 482b3 + 403b2 + 420b1 + 114045) q^39 + (193b5 + 44b4 + 159b3 + 148b2 - 3739b1 - 65900) q^41 + (-83b5 + 382b4 - 583b3 + 154b2 - 3154b1 + 60216) q^43 + (136b5 - 211b4 - 32b3 + 371b2 - 5154b1 - 110451) q^45 + (-112b5 + 84b4 - 558b3 - 557b2 - 2455b1 + 98683) q^47 + (112b5 + 371b4 - 350b3 + 322b2 - 9065b1 - 188642) q^49 + (860b5 - 449b4 + 230b3 - 560b2 - 774b1 + 396999) q^51 + (-360b5 + 258b4 + 71b3 + 777b2 - 15956b1 - 485127) q^53 + (-16b5 + 355b4 + 520b3 - 1271b2 + 1124b1 + 299751) q^55 + (6859b1 - 34295) q^57 + (787b5 + 142b4 - 747b3 - 1848b2 - 6048b1 + 489787) q^59 + (-12b5 + 541b4 - 662b3 - 243b2 - 23082b1 - 786021) q^61 + (-350b5 - 385b4 + 700b3 - 1645b2 - 4612b1 + 355301) q^63 + (-1230b5 + 451b4 - 1398b3 - 2810b2 - 20365b1 - 1215730) q^65 + (-360b5 + 859b4 + 430b3 + 96b2 + 2922b1 + 731081) q^67 + (15b5 - 306b4 + 618b3 - 1509b2 + 2709b1 - 949941) q^69 + (-3090b5 + 1417b4 - 1454b3 - 2928b2 - 4047b1 + 125414) q^71 + (2042b5 + 20b4 + 2990b3 - 5784b2 - 12686b1 - 1215727) q^73 + (1855b5 + 737b4 + 1879b3 + 3452b2 - 6233b1 + 1499959) q^75 + (-662b5 - 3407b4 + 3202b3 - 297b2 + 7040b1 - 2082317) q^77 + (-4265b5 - 3716b4 - 489b3 + 1098b2 + 14061b1 + 414154) q^79 + (-2046b5 + 2088b4 - 702b3 - 8502b2 - 214b1 - 3071005) q^81 + (354b5 - 5473b4 + 454b3 - 2504b2 - 33475b1 + 770414) q^83 + (2756b5 - 1164b4 - 1788b3 + 1729b2 + 23639b1 - 2212945) q^85 + (837b5 + 1098b4 + 2616b3 + 3645b2 - 27553b1 + 753845) q^87 + (-4939b5 - 4890b4 - 5547b3 - 3418b2 + 57567b1 - 3662084) q^89 + (1685b5 + 6830b4 - 5757b3 + 7148b2 - 13450b1 + 1462797) q^91 + (1480b5 + 6530b4 + 322b3 + 12656b2 + 72324b1 - 3117054) q^93 + (-6859b2 - 6859b1 - 48013) * q^95 + (-630b5 + 1771b4 - 1416b3 - 2224b2 + 68511b1 - 3667276) q^97 + (4717b5 - 2113b4 + 97b3 + 5798b2 - 61303b1 + 493568) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 29 q^{3} - 43 q^{5} + 908 q^{7} - 235 q^{9} - 3935 q^{11} + 6449 q^{13} - 8422 q^{15} - 6920 q^{17} + 41154 q^{19} - 54471 q^{21} + 43633 q^{23} - 48003 q^{25} - 8705 q^{27} - 134097 q^{29} + 95448 q^{31}+ \cdots + 2897895 q^{99}+O(q^{100})$ 6 * q - 29 * q^3 - 43 * q^5 + 908 * q^7 - 235 * q^9 - 3935 * q^11 + 6449 * q^13 - 8422 * q^15 - 6920 * q^17 + 41154 * q^19 - 54471 * q^21 + 43633 * q^23 - 48003 * q^25 - 8705 * q^27 - 134097 * q^29 + 95448 * q^31 - 212700 * q^33 + 202017 * q^35 - 57516 * q^37 + 683975 * q^39 - 399254 * q^41 + 359107 * q^43 - 668039 * q^45 + 590285 * q^47 - 1140196 * q^49 + 2380541 * q^51 - 2926531 * q^53 + 1799465 * q^55 - 198911 * q^57 + 2933563 * q^59 - 4738005 * q^61 + 2126109 * q^63 - 7312896 * q^65 + 4389837 * q^67 - 5697861 * q^69 + 751308 * q^71 - 7310018 * q^73 + 8992379 * q^75 - 12493471 * q^77 + 2495758 * q^79 - 18423454 * q^81 + 4583082 * q^83 - 13253407 * q^85 + 4493999 * q^87 - 21914280 * q^89 + 8775919 * q^91 - 18623792 * q^93 - 294937 * q^95 - 21931958 * q^97 + 2897895 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - x^{5} - 6373x^{4} - 12403x^{3} + 11165936x^{2} + 51537728x - 4683020288$ x^6 - x^5 - 6373*x^4 - 12403*x^3 + 11165936*x^2 + 51537728*x - 4683020288 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 11\nu^{5} - 355\nu^{4} - 47023\nu^{3} + 1765319\nu^{2} + 31097480\nu - 1668447712 ) / 1844640$ (11v^5 - 355v^4 - 47023v^3 + 1765319v^2 + 31097480*v - 1668447712) / 1844640
$\beta_{3}$	$=$	$( 149\nu^{5} + 515\nu^{4} - 752737\nu^{3} + 711641\nu^{2} + 741593960\nu - 4789303168 ) / 7378560$ (149v^5 + 515v^4 - 752737v^3 + 711641v^2 + 741593960*v - 4789303168) / 7378560
$\beta_{4}$	$=$	$( -37\nu^{5} + 2525\nu^{4} + 206081\nu^{3} - 12045433\nu^{2} - 247787800\nu + 10034105984 ) / 1844640$ (-37v^5 + 2525v^4 + 206081v^3 - 12045433v^2 - 247787800*v + 10034105984) / 1844640
$\beta_{5}$	$=$	$( -11\nu^{5} + 355\nu^{4} + 55807\nu^{3} - 1589639\nu^{2} - 57669080\nu + 1217178496 ) / 210816$ (-11v^5 + 355v^4 + 55807v^3 - 1589639v^2 - 57669080*v + 1217178496) / 210816

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} + 5\beta _1 + 2124$ b5 - b4 + b3 + 2b2 + 5b1 + 2124
$\nu^{3}$	$=$	$4\beta_{5} + 20\beta_{4} - 20\beta_{3} + 170\beta_{2} + 2925\beta _1 + 8894$ 4b5 + 20b4 - 20b3 + 170b2 + 2925*b1 + 8894
$\nu^{4}$	$=$	$4445\beta_{5} - 3923\beta_{4} + 5309\beta_{3} + 7720\beta_{2} + 25231\beta _1 + 6104270$ 4445b5 - 3923b4 + 5309b3 + 7720b2 + 25231*b1 + 6104270
$\nu^{5}$	$=$	$68\beta_{5} + 119374\beta_{4} - 74644\beta_{3} + 822592\beta_{2} + 9688655\beta _1 + 45831688$ 68b5 + 119374b4 - 74644b3 + 822592b2 + 9688655*b1 + 45831688

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

−60.0909
−37.3527
−33.1137
20.8260
49.0444
61.6869

−65.0909

165.215

774.961

2049.82

1.2

−42.3527

−290.965

−951.495

−393.248

1.3

−38.1137

−17.3107

1338.47

−734.347

1.4

15.8260

353.595

−621.598

−1936.54

1.5

44.0444

148.297

341.431

−247.091

1.6

56.6869

−401.832

26.2293

1026.40

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$19$	$-1$

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	304.8.a.j		6
4.b	odd	2	1		152.8.a.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.8.a.a	✓	6	4.b	odd	2	1
304.8.a.j		6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{6} + 29T_{3}^{5} - 6023T_{3}^{4} - 137613T_{3}^{3} + 10032066T_{3}^{2} + 159095988T_{3} - 4151704248$ T3^6 + 29*T3^5 - 6023*T3^4 - 137613*T3^3 + 10032066*T3^2 + 159095988*T3 - 4151704248 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(304))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} + \cdots - 4151704248$ T^6 + 29T^5 - 6023T^4 - 137613T^3 + 10032066T^2 + 159095988*T - 4151704248
$5$	$T^{6} + \cdots - 17534371424000$ T^6 + 43T^5 - 209449T^4 + 3480833T^3 + 9947033620T^2 - 842862313600*T - 17534371424000
$7$	$T^{6} + \cdots + 54\!\cdots\!39$ T^6 - 908T^5 - 1488299T^4 + 1000409840T^3 + 457953167643T^2 - 222136466647716*T + 5494086004338639
$11$	$T^{6} + \cdots - 96\!\cdots\!96$ T^6 + 3935T^5 - 44477759T^4 - 83497691207T^3 + 528070257267762T^2 + 224918987132640204*T - 968361891640477044696
$13$	$T^{6} + \cdots + 66\!\cdots\!40$ T^6 - 6449T^5 - 199021829T^4 + 593924159625T^3 + 10390211237480764T^2 + 17802845471734768432*T + 6610823130647883160640
$17$	$T^{6} + \cdots - 15\!\cdots\!05$ T^6 + 6920T^5 - 996776019T^4 - 7929778238224T^3 + 260775675370872419T^2 + 1763657199767012554248*T - 15108399774011124305791505
$19$	$(T - 6859)^{6}$ (T - 6859)^6
$23$	$T^{6} + \cdots - 20\!\cdots\!80$ T^6 - 43633T^5 - 1550497345T^4 + 34827487927793T^3 + 1117092000601178208T^2 + 5283964347159411099648*T - 20821094355843278965140480
$29$	$T^{6} + \cdots + 25\!\cdots\!44$ T^6 + 134097T^5 - 7847417047T^4 - 982869049695013T^3 + 8211850193258733342T^2 + 1826107301902187733753812*T + 25739514676695002150438753944
$31$	$T^{6} + \cdots - 50\!\cdots\!84$ T^6 - 95448T^5 - 76972000752T^4 + 9031930810171520T^3 + 1006363862907091519488T^2 - 88204464493269586120230912*T - 5015737415291826378896428253184
$37$	$T^{6} + \cdots + 63\!\cdots\!00$ T^6 + 57516T^5 - 575769953844T^4 - 66314833694130144T^3 + 81167495918310548318832T^2 + 14574236647268117313936381120*T + 632676065017475526491176629320000
$41$	$T^{6} + \cdots + 39\!\cdots\!00$ T^6 + 399254T^5 - 142488767300T^4 - 47975352002492024T^3 + 2146928754664332805824T^2 + 1102066300918976991345907200*T + 39978649676611020188517541171200
$43$	$T^{6} + \cdots - 97\!\cdots\!60$ T^6 - 359107T^5 - 849379341225T^4 + 61900836055876551T^3 + 118686719893007845486948T^2 - 10468743197883306891669401088*T - 97955612370440120691941016616960
$47$	$T^{6} + \cdots + 15\!\cdots\!44$ T^6 - 590285T^5 - 659030549253T^4 + 421573786509362081T^3 + 31135157812503821966040T^2 - 30948779240483471067400740864*T + 1537701602313905723710663631161344
$53$	$T^{6} + \cdots + 79\!\cdots\!60$ T^6 + 2926531T^5 + 1221553028431T^4 - 1608507930480524651T^3 - 805862513788375327610344T^2 + 49777818206954266400762962112*T + 7940496211759301864660990015916160
$59$	$T^{6} + \cdots + 31\!\cdots\!84$ T^6 - 2933563T^5 - 755403658919T^4 + 5616313217779364571T^3 - 283914843204794865185006T^2 - 1896583332949048783991348082028*T + 314098845985864837597829674288104584
$61$	$T^{6} + \cdots + 22\!\cdots\!60$ T^6 + 4738005T^5 + 4697134347149T^4 - 4739746999815490637T^3 - 7847165234790353810296602T^2 - 2019374591490846459925902264956*T + 22312187545685108848586190340356760
$67$	$T^{6} + \cdots - 34\!\cdots\!00$ T^6 - 4389837T^5 + 4652537880803T^4 + 2998862383759426865T^3 - 7151287890155899101987840T^2 + 3310899082831474845762365111808*T - 343338008123842442194046838913238400
$71$	$T^{6} + \cdots + 13\!\cdots\!12$ T^6 - 751308T^5 - 18630877286452T^4 - 1655144815918108576T^3 + 88701110463155871553217904T^2 + 76440280690026871931366444706624*T + 13738078604730015907650939336396270912
$73$	$T^{6} + \cdots + 10\!\cdots\!85$ T^6 + 7310018T^5 - 8278012810777T^4 - 111051576732070592836T^3 - 60445092449588608052912961T^2 + 176512472747222992531271525805282*T + 108333242798246457285781522425240961785
$79$	$T^{6} + \cdots - 36\!\cdots\!16$ T^6 - 2495758T^5 - 70161067813292T^4 + 199695245026197736920T^3 + 464411331123849566624471968T^2 + 163756800381559547022802305322112*T - 36141620249431372835229565634365897216
$83$	$T^{6} + \cdots - 10\!\cdots\!80$ T^6 - 4583082T^5 - 74386306437580T^4 + 175578567369227736648T^3 + 1883477193657937213439081376T^2 - 507885140489341160797770359794816*T - 10675901812341350882202347314118065108480
$89$	$T^{6} + \cdots + 22\!\cdots\!00$ T^6 + 21914280T^5 - 24773482622704T^4 - 3543541558628815081600T^3 - 23608873590304614219249060864T^2 - 3177231739570116953764701745858560*T + 224669047583280126264384056796734541004800
$97$	$T^{6} + \cdots + 17\!\cdots\!32$ T^6 + 21931958T^5 + 159978278995180T^4 + 461941973165294798664T^3 + 504997319699329652766153856T^2 + 202043977320292488913258792687616*T + 17718638496621123311109491958540050432
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.6
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