LMFDB - Newform orbit 650.6.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
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:	$x^{5} - x^{4} - 601x^{3} + 1405x^{2} + 36840x - 60300$ x^5 - x^4 - 601x^3 + 1405x^2 + 36840*x - 60300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2\cdot 5$
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.

$f(q)$	$=$	$q + 4 q^{2} + (\beta_1 - 2) q^{3} + 16 q^{4} + (4 \beta_1 - 8) q^{6} + (\beta_{2} - 3 \beta_1 + 9) q^{7} + 64 q^{8} + (\beta_{3} - 6 \beta_1 + 2) q^{9} + ( - \beta_{4} - 2 \beta_{3} + \cdots - 43) q^{11}+ \cdots + (176 \beta_{4} - 124 \beta_{3} + \cdots - 88197) q^{99}+O(q^{100})$ q + 4 * q^2 + (b1 - 2) * q^3 + 16 * q^4 + (4b1 - 8) q^6 + (b2 - 3b1 + 9) q^7 + 64 * q^8 + (b3 - 6b1 + 2) q^9 + (-b4 - 2b3 - b2 + b1 - 43) q^11 + (16b1 - 32) q^12 - 169 * q^13 + (4b2 - 12b1 + 36) * q^14 + 256 * q^16 + (b4 + b3 - 8b2 + 10b1 - 122) * q^17 + (4b3 - 24b1 + 8) * q^18 + (3b4 + 2b3 - 10b2 - 64b1 - 100) * q^19 + (2b4 - 4b3 - 4b2 + 22b1 - 744) * q^21 + (-4b4 - 8b3 - 4b2 + 4b1 - 172) * q^22 + (9b4 + 12b2 - 27b1 - 120) q^23 + (64b1 - 128) q^24 - 676 * q^26 + (8b4 - 4b3 - 8b2 + 15b1 - 1058) * q^27 + (16b2 - 48b1 + 144) * q^28 + (-7b4 + 8b3 - 22b2 - 52b1 + 117) * q^29 + (-2b4 - 19b2 + 15b1 - 2169) q^31 + 1024 * q^32 + (-17b4 - 12b3 + 6b2 - 410b1 + 524) * q^33 + (4b4 + 4b3 - 32b2 + 40b1 - 488) * q^34 + (16b3 - 96b1 + 32) * q^36 + (-4b4 + 14b3 + 24b2 + 188b1 - 2580) * q^37 + (12b4 + 8b3 - 40b2 - 256b1 - 400) * q^38 + (-169b1 + 338) q^39 + (-15b4 - 10b3 + 2b2 - 226b1 - 2250) * q^41 + (8b4 - 16b3 - 16b2 + 88b1 - 2976) * q^42 + (-49b4 - 26b3 + 20b2 - 359b1 - 342) * q^43 + (-16b4 - 32b3 - 16b2 + 16b1 - 688) * q^44 + (36b4 + 48b2 - 108b1 - 480) q^46 + (-30b4 + 38b3 + 3b2 + 233b1 - 7065) * q^47 + (256b1 - 512) q^48 + (16b4 - 21b3 + 74b2 - 204b1 - 1413) * q^49 + (-9b4 + 30b3 + 38b2 - 40b1 + 2578) * q^51 - 2704 * q^52 + (81b4 - 28b3 + 68b2 - 82b1 - 1511) * q^53 + (32b4 - 16b3 - 32b2 + 60b1 - 4232) * q^54 + (64b2 - 192b1 + 576) * q^56 + (-7b4 - 20b3 + 66b2 + 304b1 - 15400) * q^57 + (-28b4 + 32b3 - 88b2 - 208b1 + 468) * q^58 + (-66b4 - 50b3 + 139b2 - 857b1 - 8779) * q^59 + (31b4 + 36b3 - 196b2 - 1782b1 + 4157) * q^61 + (-8b4 - 76b2 + 60b1 - 8676) q^62 + (-42b4 + 38b3 - 167b2 - 1239b1 + 4979) * q^63 + 4096 * q^64 + (-68b4 - 48b3 + 24b2 - 1640b1 + 2096) * q^66 + (49b4 + 20b3 + 3b2 + 617b1 - 19847) * q^67 + (16b4 + 16b3 - 128b2 + 160b1 - 1952) * q^68 + (15b4 + 51b3 + 78b2 - 918b1 - 6357) * q^69 + (32b4 + 56b3 + 60b2 - 1210b1 + 5034) * q^71 + (64b3 - 384b1 + 128) * q^72 + (169b4 + 92b3 + 206b2 + 1508b1 + 3238) * q^73 + (-16b4 + 56b3 + 96b2 + 752b1 - 10320) * q^74 + (48b4 + 32b3 - 160b2 - 1024b1 - 1600) * q^76 + (96b4 + 75b3 + 60b2 + 1482b1 - 15124) * q^77 + (-676b1 + 1352) q^78 + (65b4 + 208b3 + 180b2 - 1031b1 - 13316) * q^79 + (-56b4 - 148b3 + 176b2 - 1412b1 + 5597) * q^81 + (-60b4 - 40b3 + 8b2 - 904b1 - 9000) * q^82 + (-193b4 + 64b3 + 63b2 + 1111b1 + 2477) * q^83 + (32b4 - 64b3 - 64b2 + 352b1 - 11904) * q^84 + (-196b4 - 104b3 + 80b2 - 1436b1 - 1368) * q^86 + (27b4 - 84b3 - 74b2 + 2873b1 - 13410) * q^87 + (-64b4 - 128b3 - 64b2 + 64b1 - 2752) * q^88 + (81b4 + 4b3 - 442b2 - 1670b1 - 57096) * q^89 + (-169b2 + 507b1 - 1521) * q^91 + (144b4 + 192b2 - 432b1 - 1920) q^92 + (-36b4 + 14b3 + 48b2 - 2044b1 + 8022) * q^93 + (-120b4 + 152b3 + 12b2 + 932b1 - 28260) * q^94 + (1024b1 - 2048) q^96 + (-322b4 - 426b3 + 660b2 + 2472b1 - 16328) * q^97 + (64b4 - 84b3 + 296b2 - 816b1 - 5652) * q^98 + (176b4 - 124b3 + 77b2 + 877b1 - 88197) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 20 q^{2} - 9 q^{3} + 80 q^{4} - 36 q^{6} + 42 q^{7} + 320 q^{8} + 4 q^{9} - 213 q^{11} - 144 q^{12} - 845 q^{13} + 168 q^{14} + 1280 q^{16} - 601 q^{17} + 16 q^{18} - 567 q^{19} - 3700 q^{21} - 852 q^{22}+ \cdots - 440284 q^{99}+O(q^{100})$ 5 * q + 20 * q^2 - 9 * q^3 + 80 * q^4 - 36 * q^6 + 42 * q^7 + 320 * q^8 + 4 * q^9 - 213 * q^11 - 144 * q^12 - 845 * q^13 + 168 * q^14 + 1280 * q^16 - 601 * q^17 + 16 * q^18 - 567 * q^19 - 3700 * q^21 - 852 * q^22 - 636 * q^23 - 576 * q^24 - 3380 * q^26 - 5283 * q^27 + 672 * q^28 + 540 * q^29 - 10828 * q^31 + 5120 * q^32 + 2227 * q^33 - 2404 * q^34 + 64 * q^36 - 12708 * q^37 - 2268 * q^38 + 1521 * q^39 - 11461 * q^41 - 14800 * q^42 - 2020 * q^43 - 3408 * q^44 - 2544 * q^46 - 35062 * q^47 - 2304 * q^48 - 7285 * q^49 + 12859 * q^51 - 13520 * q^52 - 7718 * q^53 - 21132 * q^54 + 2688 * q^56 - 76689 * q^57 + 2160 * q^58 - 44686 * q^59 + 18972 * q^61 - 43312 * q^62 + 23698 * q^63 + 20480 * q^64 + 8908 * q^66 - 98667 * q^67 - 9616 * q^68 - 32718 * q^69 + 23928 * q^71 + 256 * q^72 + 17529 * q^73 - 50832 * q^74 - 9072 * q^76 - 74234 * q^77 + 6084 * q^78 - 67676 * q^79 + 26629 * q^81 - 45844 * q^82 + 13689 * q^83 - 59200 * q^84 - 8080 * q^86 - 64204 * q^87 - 13632 * q^88 - 287231 * q^89 - 7098 * q^91 - 10176 * q^92 + 38102 * q^93 - 140248 * q^94 - 9216 * q^96 - 78846 * q^97 - 29140 * q^98 - 440284 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - x^{4} - 601x^{3} + 1405x^{2} + 36840x - 60300$ x^5 - x^4 - 601*x^3 + 1405*x^2 + 36840*x - 60300 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{4} - \nu^{3} - 571\nu^{2} + 1225\nu + 22290 ) / 120$ (v^4 - v^3 - 571v^2 + 1225v + 22290) / 120
$\beta_{3}$	$=$	$\nu^{2} + 2\nu - 241$ v^2 + 2*v - 241
$\beta_{4}$	$=$	$( \nu^{4} + 14\nu^{3} - 601\nu^{2} - 5990\nu + 38160 ) / 120$ (v^4 + 14v^3 - 601v^2 - 5990*v + 38160) / 120

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} - 2\beta _1 + 241$ b3 - 2*b1 + 241
$\nu^{3}$	$=$	$8\beta_{4} + 2\beta_{3} - 8\beta_{2} + 477\beta _1 - 576$ 8b4 + 2b3 - 8b2 + 477b1 - 576
$\nu^{4}$	$=$	$8\beta_{4} + 573\beta_{3} + 112\beta_{2} - 1890\beta _1 + 114745$ 8b4 + 573b3 + 112b2 - 1890b1 + 114745

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

−23.8063
−7.92606
1.60591
8.98932
22.1371

4.00000

−25.8063

16.0000

−103.225

115.455

64.0000

422.964

1.2

4.00000

−9.92606

16.0000

−39.7042

−124.276

64.0000

−144.473

1.3

4.00000

−0.394093

16.0000

−1.57637

194.075

64.0000

−242.845

1.4

4.00000

6.98932

16.0000

27.9573

−76.6003

64.0000

−194.149

1.5

4.00000

20.1371

16.0000

80.5485

−66.6546

64.0000

162.503

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$13$	$+1$

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	650.6.a.p	yes	5
5.b	even	2	1		650.6.a.o	✓	5
5.c	odd	4	2		650.6.b.m		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
650.6.a.o	✓	5	5.b	even	2	1
650.6.a.p	yes	5	1.a	even	1	1	trivial
650.6.b.m		10	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{5} + 9T_{3}^{4} - 569T_{3}^{3} - 2145T_{3}^{2} + 35296T_{3} + 14208$ T3^5 + 9*T3^4 - 569*T3^3 - 2145*T3^2 + 35296*T3 + 14208 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(650))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 4)^{5}$ (T - 4)^5
$3$	$T^{5} + 9 T^{4} + \cdots + 14208$ T^5 + 9T^4 - 569T^3 - 2145T^2 + 35296T + 14208
$5$	$T^{5}$ T^5
$7$	$T^{5} + \cdots + 14217728944$ T^5 - 42T^4 - 37493T^3 - 461912T^2 + 316914732T + 14217728944
$11$	$T^{5} + \cdots - 310309345876$ T^5 + 213T^4 - 519791T^3 - 124322887T^2 + 24495434940T - 310309345876
$13$	$(T + 169)^{5}$ (T + 169)^5
$17$	$T^{5} + \cdots + 59701212884156$ T^5 + 601T^4 - 2219653T^3 - 548023313T^2 + 555791815936T + 59701212884156
$19$	$T^{5} + \cdots + 57\!\cdots\!32$ T^5 + 567T^4 - 7263860T^3 - 7410838788T^2 + 6036235891248T + 5749505909730432
$23$	$T^{5} + \cdots + 20\!\cdots\!00$ T^5 + 636T^4 - 21164661T^3 - 4889964276T^2 + 38586539405700T + 20308915254420000
$29$	$T^{5} + \cdots - 21\!\cdots\!76$ T^5 - 540T^4 - 42742958T^3 + 134499464776T^2 - 89367020511003T - 21337462504523876
$31$	$T^{5} + \cdots + 190703824040808$ T^5 + 10828T^4 + 34169743T^3 + 34877323974T^2 + 5410973728596T + 190703824040808
$37$	$T^{5} + \cdots - 46\!\cdots\!12$ T^5 + 12708T^4 - 12726668T^3 - 304046735104T^2 - 704971488778944T - 469525976468941312
$41$	$T^{5} + \cdots + 17\!\cdots\!00$ T^5 + 11461T^4 - 17412760T^3 - 292906821760T^2 + 53134267904000T + 1757701439905792000
$43$	$T^{5} + \cdots + 11\!\cdots\!80$ T^5 + 2020T^4 - 499606993T^3 - 1156957805312T^2 + 57027546975197380T + 117620382030661807280
$47$	$T^{5} + \cdots - 34\!\cdots\!48$ T^5 + 35062T^4 - 63411953T^3 - 6074792218536T^2 + 33659280051730896T - 34116647855556267648
$53$	$T^{5} + \cdots + 71\!\cdots\!50$ T^5 + 7718T^4 - 1833017230T^3 - 16501332771944T^2 + 724258497589713725T + 7134060570679624382450
$59$	$T^{5} + \cdots - 35\!\cdots\!60$ T^5 + 44686T^4 - 1064382961T^3 - 29517555480196T^2 + 709438621920308240T - 3535853193885725812160
$61$	$T^{5} + \cdots + 33\!\cdots\!08$ T^5 - 18972T^4 - 3270109158T^3 + 9932162118436T^2 + 2895841011979547541T + 33638376569332128637608
$67$	$T^{5} + \cdots - 17\!\cdots\!04$ T^5 + 98667T^4 + 3258386929T^3 + 40345560200727T^2 + 150500719022582460T - 171013190408966294604
$71$	$T^{5} + \cdots - 36\!\cdots\!40$ T^5 - 23928T^4 - 1208459180T^3 + 13512741542864T^2 + 24454249216197120T - 360349430843365027840
$73$	$T^{5} + \cdots - 50\!\cdots\!00$ T^5 - 17529T^4 - 7494628596T^3 + 122438806092080T^2 + 11388019292946027840T - 50084448943677342807600
$79$	$T^{5} + \cdots + 99\!\cdots\!56$ T^5 + 67676T^4 - 5098454125T^3 - 254810881736876T^2 + 4080795021410167268T + 99123818966338926777056
$83$	$T^{5} + \cdots - 52\!\cdots\!00$ T^5 - 13689T^4 - 10407726851T^3 + 318951000873975T^2 + 15426159035422854000T - 528676498930753497600000
$89$	$T^{5} + \cdots - 23\!\cdots\!44$ T^5 + 287231T^4 + 23611514768T^3 - 76280430883972T^2 - 81027768216151092704T - 2394542958821076856074944
$97$	$T^{5} + \cdots + 37\!\cdots\!00$ T^5 + 78846T^4 - 44314935620T^3 - 3333415397905112T^2 + 466257955168109112960T + 37746196640579399377206400
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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 1.5
Significant digits:
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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	650.6.a.p

Level	$650$
Weight	$6$
Character orbit	650.a
Self dual	yes
Analytic conductor	$104.249$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$