LMFDB - Newform orbit 13.6.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [13,6,Mod(1,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("13.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	13.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.08498965757$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.168897.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 100x + 256$ x^3 - x^2 - 100*x + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
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 $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 + 2) q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{3} + (4 \beta_{2} + \beta_1 + 40) q^{4} + ( - \beta_{2} - 7 \beta_1 + 21) q^{5} + ( - 10 \beta_{2} - \beta_1 - 66) q^{6} + ( - 3 \beta_{2} - 3 \beta_1 - 19) q^{7}+ \cdots + ( - 1928 \beta_{2} + 4720 \beta_1 - 21684) q^{99}+O(q^{100})$ q + (b1 + 2) * q^2 + (-b2 - b1 + 3) * q^3 + (4b2 + b1 + 40) q^4 + (-b2 - 7b1 + 21) q^5 + (-10b2 - b1 - 66) q^6 + (-3b2 - 3b1 - 19) * q^7 + (28b2 + 27b1 + 100) * q^8 + (b2 + 7b1 - 66) q^9 + (-34b2 + 23b1 - 438) * q^10 + (30b2 - 26b1 + 194) * q^11 + (-32b2 - 83b1 - 336) * q^12 + 169 * q^13 + (-30b2 - 31b1 - 254) * q^14 + (31b2 - 17b1 + 663) * q^15 + (148b2 + 181b1 + 868) * q^16 + (-93b2 + 77b1 + 277) * q^17 + (34b2 - 68b1 + 348) * q^18 + (-134b2 + 178b1 - 10) * q^19 + (-80b2 - 407b1 - 120) * q^20 + (31b2 + 49b1 + 447) * q^21 + (76b2 + 370b1 - 1260) * q^22 + (152b2 - 72b1 + 1232) * q^23 + (-204b2 - 381b1 - 4332) * q^24 + (205b2 - 365b1 + 796) * q^25 + (169b1 + 338) q^26 + (257b2 + 305b1 - 1527) * q^27 + (-208b2 - 277b1 - 2128) * q^28 + (-584b2 + 56b1 - 2938) * q^29 + (118b2 + 835b1 + 294) * q^30 + (-268b2 - 148b1 - 820) * q^31 + (716b2 + 563b1 + 11436) * q^32 + (134b2 - 550b1 - 426) * q^33 + (-250b2 - 265b1 + 5418) * q^34 + (121b2 + 145b1 + 1401) * q^35 + (-100b2 + 362b1 - 1680) * q^36 + (419b2 - 451b1 - 6727) * q^37 + (-92b2 - 858b1 + 11548) * q^38 + (-169b2 - 169b1 + 507) * q^39 + (-1020b2 - 849b1 - 14220) * q^40 + (-250b2 + 858b1 - 3952) * q^41 + (382b2 + 553b1 + 4350) * q^42 + (697b2 - 431b1 + 821) * q^43 + (976b2 - 418b1 + 16736) * q^44 + (-160b2 + 680b1 - 4866) * q^45 + (624b2 + 2064b1 - 1824) * q^46 + (-807b2 + 33b1 + 11409) * q^47 + (-1724b2 - 2315b1 - 24636) * q^48 + (177b2 + 231b1 - 14934) * q^49 + (-230b2 + 2186b1 - 22408) * q^50 + (-1265b2 + 823b1 + 4215) * q^51 + (676b2 + 169b1 + 6760) * q^52 + (2154b2 + 822b1 - 4464) * q^53 + (2762b2 - 547b1 + 18714) * q^54 + (578b2 - 3550b1 + 12954) * q^55 + (-1396b2 - 1899b1 - 15796) * q^56 + (-1950b2 + 1662b1 + 18) * q^57 + (-3280b2 - 5914b1 - 4404) * q^58 + (-950b2 + 1458b1 + 20830) * q^59 + (3056b2 + 593b1 + 36624) * q^60 + (2330b2 + 1910b1 - 4888) * q^61 + (-2200b2 - 2012b1 - 12776) * q^62 + (14b2 - 10b1 - 546) * q^63 + (1812b2 + 8661b1 + 36244) * q^64 + (-169b2 - 1183b1 + 3549) * q^65 + (-1396b2 + 794b1 - 37716) * q^66 + (-1666b2 + 1478b1 + 18218) * q^67 + (416b2 + 1969b1 - 17048) * q^68 + (-48b2 - 2976b1 - 5712) * q^69 + (1306b2 + 1861b1 + 13146) * q^70 + (3127b2 - 633b1 + 26071) * q^71 + (-240b2 - 366b1 + 9720) * q^72 + (-2568b2 - 2712b1 - 13438) * q^73 + (710b2 - 4181b1 - 42446) * q^74 + (2944b2 - 3416b1 + 8988) * q^75 + (304b2 + 6250b1 - 35296) * q^76 + (-438b2 - 922b1 - 6710) * q^77 + (-1690b2 - 169b1 - 11154) * q^78 + (-3576b2 - 2856b1 - 19672) * q^79 + (-6956b2 - 5447b1 - 86412) * q^80 + (-128b2 - 2696b1 - 35175) * q^81 + (1932b2 - 6060b1 + 49440) * q^82 + (4840b2 + 3808b1 + 31428) * q^83 + (3512b2 + 4139b1 + 33528) * q^84 + (-2555b2 + 4723b1 - 19983) * q^85 + (2458b2 + 4737b1 - 24878) * q^86 + (154b2 + 9418b1 + 43218) * q^87 + (1752b2 + 10194b1 + 49272) * q^88 + (-432b2 - 8864b1 + 14186) * q^89 + (1760b2 - 6346b1 + 35868) * q^90 + (-507b2 - 507b1 - 3211) * q^91 + (7136b2 + 1536b1 + 99776) * q^92 + (932b2 + 3620b1 + 33924) * q^93 + (-4710b2 + 7341b1 + 21834) * q^94 + (-5418b2 + 12150b1 - 69570) * q^95 + (-13076b2 - 18749b1 - 74964) * q^96 + (500b2 - 1396b1 + 25910) * q^97 + (1986b2 - 14280b1 - 13452) * q^98 + (-1928b2 + 4720b1 - 21684) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 7 q^{2} + 8 q^{3} + 121 q^{4} + 56 q^{5} - 199 q^{6} - 60 q^{7} + 327 q^{8} - 191 q^{9} - 1291 q^{10} + 556 q^{11} - 1091 q^{12} + 507 q^{13} - 793 q^{14} + 1972 q^{15} + 2785 q^{16} + 908 q^{17}+ \cdots - 60332 q^{99}+O(q^{100})$ 3 * q + 7 * q^2 + 8 * q^3 + 121 * q^4 + 56 * q^5 - 199 * q^6 - 60 * q^7 + 327 * q^8 - 191 * q^9 - 1291 * q^10 + 556 * q^11 - 1091 * q^12 + 507 * q^13 - 793 * q^14 + 1972 * q^15 + 2785 * q^16 + 908 * q^17 + 976 * q^18 + 148 * q^19 - 767 * q^20 + 1390 * q^21 - 3410 * q^22 + 3624 * q^23 - 13377 * q^24 + 2023 * q^25 + 1183 * q^26 - 4276 * q^27 - 6661 * q^28 - 8758 * q^29 + 1717 * q^30 - 2608 * q^31 + 34871 * q^32 - 1828 * q^33 + 15989 * q^34 + 4348 * q^35 - 4678 * q^36 - 20632 * q^37 + 33786 * q^38 + 1352 * q^39 - 43509 * q^40 - 10998 * q^41 + 13603 * q^42 + 2032 * q^43 + 49790 * q^44 - 13918 * q^45 - 3408 * q^46 + 34260 * q^47 - 76223 * q^48 - 44571 * q^49 - 65038 * q^50 + 13468 * q^51 + 20449 * q^52 - 12570 * q^53 + 55595 * q^54 + 35312 * q^55 - 49287 * q^56 + 1716 * q^57 - 19126 * q^58 + 63948 * q^59 + 110465 * q^60 - 12754 * q^61 - 40340 * q^62 - 1648 * q^63 + 117393 * q^64 + 9464 * q^65 - 112354 * q^66 + 56132 * q^67 - 49175 * q^68 - 20112 * q^69 + 41299 * q^70 + 77580 * q^71 + 28794 * q^72 - 43026 * q^73 - 131519 * q^74 + 23548 * q^75 - 99638 * q^76 - 21052 * q^77 - 33631 * q^78 - 61872 * q^79 - 264683 * q^80 - 108221 * q^81 + 142260 * q^82 + 98092 * q^83 + 104723 * q^84 - 55226 * q^85 - 69897 * q^86 + 139072 * q^87 + 158010 * q^88 + 33694 * q^89 + 101258 * q^90 - 10140 * q^91 + 300864 * q^92 + 105392 * q^93 + 72843 * q^94 - 196560 * q^95 - 243641 * q^96 + 76334 * q^97 - 54636 * q^98 - 60332 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 100x + 256$ x^3 - x^2 - 100*x + 256 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} + 3\nu - 68 ) / 4$ (v^2 + 3*v - 68) / 4

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$4\beta_{2} - 3\beta _1 + 68$ 4b2 - 3b1 + 68

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

−10.6486
2.68079
8.96778

−8.64858

10.2870

42.7979

92.1784

−88.9676

2.86088

−93.3863

−137.178

−797.212

1.2

4.68079

13.5120

−10.0902

15.4272

63.2466

12.5359

−197.015

−60.4272

72.2115

1.3

10.9678

−15.7989

88.2923

−51.6056

−173.279

−75.3967

617.402

6.60562

−565.999

Atkin-Lehner signs

$p$	Sign
$13$	$-1$

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	13.6.a.b	✓	3
3.b	odd	2	1		117.6.a.d		3
4.b	odd	2	1		208.6.a.j		3
5.b	even	2	1		325.6.a.c		3
5.c	odd	4	2		325.6.b.c		6
7.b	odd	2	1		637.6.a.b		3
8.b	even	2	1		832.6.a.s		3
8.d	odd	2	1		832.6.a.t		3
13.b	even	2	1		169.6.a.b		3
13.d	odd	4	2		169.6.b.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.6.a.b	✓	3	1.a	even	1	1	trivial
117.6.a.d		3	3.b	odd	2	1
169.6.a.b		3	13.b	even	2	1
169.6.b.b		6	13.d	odd	4	2
208.6.a.j		3	4.b	odd	2	1
325.6.a.c		3	5.b	even	2	1
325.6.b.c		6	5.c	odd	4	2
637.6.a.b		3	7.b	odd	2	1
832.6.a.s		3	8.b	even	2	1
832.6.a.t		3	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{3} - 7T_{2}^{2} - 84T_{2} + 444$ T2^3 - 7*T2^2 - 84*T2 + 444 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(13))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} - 7 T^{2} + \cdots + 444$ T^3 - 7T^2 - 84T + 444
$3$	$T^{3} - 8 T^{2} + \cdots + 2196$ T^3 - 8T^2 - 237T + 2196
$5$	$T^{3} - 56 T^{2} + \cdots + 73386$ T^3 - 56T^2 - 4131T + 73386
$7$	$T^{3} + 60 T^{2} + \cdots + 2704$ T^3 + 60T^2 - 1125T + 2704
$11$	$T^{3} - 556 T^{2} + \cdots + 39698256$ T^3 - 556T^2 - 78420T + 39698256
$13$	$(T - 169)^{3}$ (T - 169)^3
$17$	$T^{3} - 908 T^{2} + \cdots + 77884638$ T^3 - 908T^2 - 1417827T + 77884638
$19$	$T^{3} + \cdots + 1415854512$ T^3 - 148T^2 - 5297972T + 1415854512
$23$	$T^{3} + \cdots + 5045833728$ T^3 - 3624T^2 + 786048T + 5045833728
$29$	$T^{3} + \cdots - 221025174456$ T^3 + 8758T^2 - 22280052T - 221025174456
$31$	$T^{3} + \cdots - 1607044480$ T^3 + 2608T^2 - 10731952T - 1607044480
$37$	$T^{3} + \cdots + 46212896426$ T^3 + 20632T^2 + 99943613T + 46212896426
$41$	$T^{3} + \cdots + 29456898048$ T^3 + 10998T^2 - 38709120T + 29456898048
$43$	$T^{3} + \cdots + 281385762060$ T^3 - 2032T^2 - 80653829T + 281385762060
$47$	$T^{3} + \cdots - 696870885384$ T^3 - 34260T^2 + 299766555T - 696870885384
$53$	$T^{3} + \cdots - 4415410372608$ T^3 + 12570T^2 - 699425280T - 4415410372608
$59$	$T^{3} + \cdots - 1932677407728$ T^3 - 63948T^2 + 1046124780T - 1932677407728
$61$	$T^{3} + \cdots - 18650455523968$ T^3 + 12754T^2 - 1152934528T - 18650455523968
$67$	$T^{3} + \cdots + 2080268535536$ T^3 - 56132T^2 + 481597100T + 2080268535536
$71$	$T^{3} + \cdots + 37395101110464$ T^3 - 77580T^2 + 620949795T + 37395101110464
$73$	$T^{3} + \cdots + 5649650834008$ T^3 + 43026T^2 - 1169104212T + 5649650834008
$79$	$T^{3} + \cdots - 2044988893184$ T^3 + 61872T^2 - 1519042752T - 2044988893184
$83$	$T^{3} + \cdots + 17971240920768$ T^3 - 98092T^2 - 1863905040T + 17971240920768
$89$	$T^{3} + \cdots - 28887869991912$ T^3 - 33694T^2 - 7596222420T - 28887869991912
$97$	$T^{3} + \cdots - 12102379894216$ T^3 - 76334T^2 + 1723377980T - 12102379894216
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