LMFDB - Newform orbit 28.10.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,10,Mod(1,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("28.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$28 = 2^{2} \cdot 7$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	28.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:	$14.4210034126$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4561})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1140$ x^2 - x - 1140
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}$
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 $\beta = 2\sqrt{4561}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta - 112) q^{3} + (9 \beta + 798) q^{5} + 2401 q^{7} + (224 \beta + 11105) q^{9} + ( - 168 \beta - 25296) q^{11} + ( - 585 \beta - 92386) q^{13} + ( - 1806 \beta - 253572) q^{15} + (3042 \beta - 76062) q^{17}+ \cdots + ( - 7531944 \beta - 967470288) q^{99}+O(q^{100})$ q + (-b - 112) * q^3 + (9b + 798) q^5 + 2401 * q^7 + (224b + 11105) q^9 + (-168b - 25296) q^11 + (-585b - 92386) q^13 + (-1806b - 253572) q^15 + (3042b - 76062) q^17 + (2097b - 302512) q^19 + (-2401b - 268912) q^21 + (-546b - 1251804) q^23 + (14364b + 161443) q^25 + (-16510b - 3125920) q^27 + (1806b - 1363746) q^29 + (-44226b - 3327520) q^31 + (44112b + 5898144) q^33 + (21609b + 1915998) q^35 + (-99918b - 3430114) q^37 + (157906b + 21019972) q^39 + (-59694b + 7077378) q^41 + (-116424b - 566920) q^43 + (278697b + 45641694) q^45 + (68286b + 17743320) q^47 + 5764801 * q^49 + (-264642b - 46979304) q^51 + (-486780b - 16655970) q^53 + (-361728b - 47771136) q^55 + (67648b - 4376324) q^57 + (858429b + 14709744) q^59 + (577161b + 17975846) q^61 + (537824b + 26663105) q^63 + (-1298304b - 169778688) q^65 + (548730b - 85143352) q^67 + (1312956b + 150163272) q^69 + (-29568b - 231807744) q^71 + (-1998828b - 50403430) q^73 + (-1770211b - 280138432) q^75 + (-403368b - 60735696) q^77 + (1018836b - 98746912) q^79 + (566048b + 432731765) q^81 + (-1920243b + 25713744) q^83 + (1742958b + 438786756) q^85 + (1161474b + 119790888) q^87 + (1535628b + 324643746) q^89 + (-1404585b - 221818786) q^91 + (8280832b + 1179541384) q^93 + (-1049202b + 102914436) q^95 + (-1516194b + 1390646810) q^97 + (-7531944b - 967470288) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 224 q^{3} + 1596 q^{5} + 4802 q^{7} + 22210 q^{9} - 50592 q^{11} - 184772 q^{13} - 507144 q^{15} - 152124 q^{17} - 605024 q^{19} - 537824 q^{21} - 2503608 q^{23} + 322886 q^{25} - 6251840 q^{27} - 2727492 q^{29}+ \cdots - 1934940576 q^{99}+O(q^{100})$ 2 * q - 224 * q^3 + 1596 * q^5 + 4802 * q^7 + 22210 * q^9 - 50592 * q^11 - 184772 * q^13 - 507144 * q^15 - 152124 * q^17 - 605024 * q^19 - 537824 * q^21 - 2503608 * q^23 + 322886 * q^25 - 6251840 * q^27 - 2727492 * q^29 - 6655040 * q^31 + 11796288 * q^33 + 3831996 * q^35 - 6860228 * q^37 + 42039944 * q^39 + 14154756 * q^41 - 1133840 * q^43 + 91283388 * q^45 + 35486640 * q^47 + 11529602 * q^49 - 93958608 * q^51 - 33311940 * q^53 - 95542272 * q^55 - 8752648 * q^57 + 29419488 * q^59 + 35951692 * q^61 + 53326210 * q^63 - 339557376 * q^65 - 170286704 * q^67 + 300326544 * q^69 - 463615488 * q^71 - 100806860 * q^73 - 560276864 * q^75 - 121471392 * q^77 - 197493824 * q^79 + 865463530 * q^81 + 51427488 * q^83 + 877573512 * q^85 + 239581776 * q^87 + 649287492 * q^89 - 443637572 * q^91 + 2359082768 * q^93 + 205828872 * q^95 + 2781293620 * q^97 - 1934940576 * q^99

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

34.2676
−33.2676

−247.070

2013.63

2401.00

41360.8

1.2

23.0704

−417.633

2401.00

−19150.8

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$-1$

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	28.10.a.a	✓	2
3.b	odd	2	1		252.10.a.a		2
4.b	odd	2	1		112.10.a.f		2
7.b	odd	2	1		196.10.a.c		2
7.c	even	3	2		196.10.e.f		4
7.d	odd	6	2		196.10.e.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.10.a.a	✓	2	1.a	even	1	1	trivial
112.10.a.f		2	4.b	odd	2	1
196.10.a.c		2	7.b	odd	2	1
196.10.e.c		4	7.d	odd	6	2
196.10.e.f		4	7.c	even	3	2
252.10.a.a		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 224T_{3} - 5700$ T3^2 + 224*T3 - 5700 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(28))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 224T - 5700$ T^2 + 224*T - 5700
$5$	$T^{2} - 1596 T - 840960$ T^2 - 1596*T - 840960
$7$	$(T - 2401)^{2}$ (T - 2401)^2
$11$	$T^{2} + 50592 T + 124968960$ T^2 + 50592*T + 124968960
$13$	$T^{2} + \cdots + 2291620096$ T^2 + 184772*T + 2291620096
$17$	$T^{2} + \cdots - 163040242572$ T^2 + 152124*T - 163040242572
$19$	$T^{2} + \cdots + 11287180348$ T^2 + 605024*T + 11287180348
$23$	$T^{2} + \cdots + 1561574426112$ T^2 + 2503608*T + 1561574426112
$29$	$T^{2} + \cdots + 1800297865332$ T^2 + 2727492*T + 1800297865332
$31$	$T^{2} + \cdots - 24611763152144$ T^2 + 6655040*T - 24611763152144
$37$	$T^{2} + \cdots - 170375239019660$ T^2 + 6860228*T - 170375239019660
$41$	$T^{2} + \cdots - 14920909260300$ T^2 - 14154756*T - 14920909260300
$43$	$T^{2} + \cdots - 246967771338944$ T^2 + 1133840*T - 246967771338944
$47$	$T^{2} + \cdots + 229754037712176$ T^2 - 35486640*T + 229754037712176
$53$	$T^{2} + \cdots - 40\!\cdots\!00$ T^2 + 33311940*T - 4045581458048700
$59$	$T^{2} + \cdots - 13\!\cdots\!68$ T^2 - 29419488*T - 13227633381114468
$61$	$T^{2} + \cdots - 57\!\cdots\!08$ T^2 - 35951692*T - 5754215735223008
$67$	$T^{2} + \cdots + 17\!\cdots\!04$ T^2 + 170286704*T + 1756037832048304
$71$	$T^{2} + \cdots + 53\!\cdots\!80$ T^2 + 463615488*T + 53718880058081280
$73$	$T^{2} + \cdots - 70\!\cdots\!96$ T^2 + 100806860*T - 70349991431901596
$79$	$T^{2} + \cdots - 91\!\cdots\!80$ T^2 + 197493824*T - 9186808216546880
$83$	$T^{2} + \cdots - 66\!\cdots\!20$ T^2 - 51427488*T - 66610509888072420
$89$	$T^{2} + \cdots + 62\!\cdots\!20$ T^2 - 649287492*T + 62371412019530820
$97$	$T^{2} + \cdots + 18\!\cdots\!16$ T^2 - 2781293620*T + 1891958435745792916
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