LMFDB - Newform orbit 196.10.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,10,Mod(165,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("196.165");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$196 = 2^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	196.e (of order $3$ , degree $2$ , not minimal)

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:	no
Analytic conductor:	$100.947023888$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{4561})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 1141x^{2} + 1140x + 1299600$ x^4 - x^3 + 1141x^2 + 1140x + 1299600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$ .

$n$	$99$	$101$
$\chi(n)$	$1$	$-1 + \beta_{1}$

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}

165.1

−16.6338	+	28.8106i
17.1338	−	29.6766i
−16.6338	−	28.8106i
17.1338	+	29.6766i

−123.535

213.969i

1006.82

1743.86i

−20680.4

−

35819.5i

165.2

11.5352

−

19.9795i

−208.817

−

361.681i

9575.38

16585.0i

177.1

−123.535

−

213.969i

1006.82

−

1743.86i

−20680.4

35819.5i

177.2

11.5352

19.9795i

−208.817

361.681i

9575.38

−

16585.0i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.10.e.c		4
7.b	odd	2	1		196.10.e.f		4
7.c	even	3	1		196.10.a.c		2
7.c	even	3	1	inner	196.10.e.c		4
7.d	odd	6	1		28.10.a.a	✓	2
7.d	odd	6	1		196.10.e.f		4
21.g	even	6	1		252.10.a.a		2
28.f	even	6	1		112.10.a.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.10.a.a	✓	2	7.d	odd	6	1
112.10.a.f		2	28.f	even	6	1
196.10.a.c		2	7.c	even	3	1
196.10.e.c		4	1.a	even	1	1	trivial
196.10.e.c		4	7.c	even	3	1	inner
196.10.e.f		4	7.b	odd	2	1
196.10.e.f		4	7.d	odd	6	1
252.10.a.a		2	21.g	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} + 224T_{3}^{3} + 55876T_{3}^{2} - 1276800T_{3} + 32490000$ T3^4 + 224*T3^3 + 55876*T3^2 - 1276800*T3 + 32490000 acting on $S_{10}^{\mathrm{new}}(196, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 224 T^{3} + \cdots + 32490000$ T^4 + 224T^3 + 55876T^2 - 1276800*T + 32490000
$5$	$T^{4} + \cdots + 707213721600$ T^4 - 1596T^3 + 3388176T^2 + 1342172160*T + 707213721600
$7$	$T^{4}$ T^4
$11$	$T^{4} + \cdots + 15\!\cdots\!00$ T^4 - 50592T^3 + 2434581504T^2 - 6322429624320*T + 15617240963481600
$13$	$(T^{2} - 184772 T + 2291620096)^{2}$ (T^2 - 184772*T + 2291620096)^2
$17$	$T^{4} + \cdots + 26\!\cdots\!84$ T^4 + 152124T^3 + 186181953948T^2 - 24802333861022928*T + 26582120697936601175184
$19$	$T^{4} + \cdots + 12\!\cdots\!04$ T^4 + 605024T^3 + 354766860228T^2 + 6829015002868352*T + 127400440208277401104
$23$	$T^{4} + \cdots + 24\!\cdots\!44$ T^4 - 2503608T^3 + 4706478591552T^2 - 3909570225809412096*T + 2438514688287022147436544
$29$	$(T^{2} + \cdots + 1800297865332)^{2}$ (T^2 + 2727492*T + 1800297865332)^2
$31$	$T^{4} + \cdots + 60\!\cdots\!36$ T^4 + 6655040T^3 + 68901320553744T^2 - 163792268248044405760*T + 605738885457233162891796736
$37$	$T^{4} + \cdots + 29\!\cdots\!00$ T^4 - 6860228T^3 + 217437967231644T^2 + 1168812985229364082480*T + 29027722071006275397866515600
$41$	$(T^{2} + \cdots - 14920909260300)^{2}$ (T^2 + 14154756*T - 14920909260300)^2
$43$	$(T^{2} + \cdots - 246967771338944)^{2}$ (T^2 + 1133840*T - 246967771338944)^2
$47$	$T^{4} + \cdots + 52\!\cdots\!76$ T^4 - 35486640T^3 + 1029547580777424T^2 - 8153198824838413328640*T + 52786917845047991616218654976
$53$	$T^{4} + \cdots + 16\!\cdots\!00$ T^4 - 33311940T^3 + 5155266804612300T^2 + 134766166795630811478000*T + 16366729333707445398011571690000
$59$	$T^{4} + \cdots + 17\!\cdots\!24$ T^4 - 29419488T^3 + 14093139655296612T^2 + 389150201524096517952384*T + 174970284865173772636725718923024
$61$	$T^{4} + \cdots + 33\!\cdots\!64$ T^4 - 35951692T^3 + 7046739892885872T^2 + 206873791814291134929536*T + 33110998727488062510311492568064
$67$	$T^{4} + \cdots + 30\!\cdots\!16$ T^4 - 170286704T^3 + 27241523727135312T^2 - 299029894518811256950016*T + 3083668867584907526876189276416
$71$	$(T^{2} + \cdots + 53\!\cdots\!80)^{2}$ (T^2 + 463615488*T + 53718880058081280)^2
$73$	$T^{4} + \cdots + 49\!\cdots\!16$ T^4 + 100806860T^3 + 80512014454961196T^2 - 7091761737276903721748560*T + 4949121294468627969510260627347216
$79$	$T^{4} + \cdots + 84\!\cdots\!00$ T^4 - 197493824T^3 + 48190618734689856T^2 + 1814337885040463406469120*T + 84397445207613266010631237734400
$83$	$(T^{2} + \cdots - 66\!\cdots\!20)^{2}$ (T^2 + 51427488*T - 66610509888072420)^2
$89$	$T^{4} + \cdots + 38\!\cdots\!00$ T^4 - 649287492T^3 + 359202835248119244T^2 - 40496977682659821134503440*T + 3890193037310073642217132929872400
$97$	$(T^{2} + \cdots + 18\!\cdots\!16)^{2}$ (T^2 + 2781293620*T + 1891958435745792916)^2
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Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + (\beta_{2} - 112 \beta_1) q^{3} + (9 \beta_{3} + 9 \beta_{2} + \cdots + 798) q^{5} + ( - 224 \beta_{3} - 224 \beta_{2} + \cdots - 11105) q^{9} + ( - 168 \beta_{2} + 25296 \beta_1) q^{11} + (585 \beta_{3} + 92386) q^{13}+ \cdots + ( - 7531944 \beta_{3} - 967470288) q^{99}+O(q^{100})$ q + (b2 - 112b1) q^3 + (9b3 + 9b2 - 798b1 + 798) q^5 + (-224b3 - 224b2 + 11105b1 - 11105) q^9 + (-168b2 + 25296b1) * q^11 + (585b3 + 92386) q^13 + (-1806b3 - 253572) q^15 + (-3042b2 - 76062b1) * q^17 + (2097b3 + 2097b2 + 302512b1 - 302512) q^19 + (546b3 + 546b2 - 1251804b1 + 1251804) q^23 + (14364b2 - 161443b1) * q^25 + (16510b3 + 3125920) q^27 + (1806b3 - 1363746) q^29 + (44226b2 - 3327520b1) * q^31 + (44112b3 + 44112b2 - 5898144b1 + 5898144) q^33 + (99918b3 + 99918b2 - 3430114b1 + 3430114) q^37 + (157906b2 - 21019972b1) * q^39 + (59694b3 - 7077378) q^41 + (-116424b3 - 566920) q^43 + (-278697b2 + 45641694b1) * q^45 + (68286b3 + 68286b2 - 17743320b1 + 17743320) q^47 + (264642b3 + 264642b2 - 46979304b1 + 46979304) q^51 + (-486780b2 + 16655970b1) * q^53 + (361728b3 + 47771136) q^55 + (67648b3 - 4376324) q^57 + (-858429b2 + 14709744b1) * q^59 + (577161b3 + 577161b2 - 17975846b1 + 17975846) q^61 + (1298304b3 + 1298304b2 - 169778688b1 + 169778688) q^65 + (548730b2 + 85143352b1) * q^67 + (-1312956b3 - 150163272) q^69 + (-29568b3 - 231807744) q^71 + (1998828b2 - 50403430b1) * q^73 + (-1770211b3 - 1770211b2 + 280138432b1 - 280138432) q^75 + (-1018836b3 - 1018836b2 - 98746912b1 + 98746912) q^79 + (566048b2 - 432731765b1) * q^81 + (1920243b3 - 25713744) q^83 + (1742958b3 + 438786756) q^85 + (-1161474b2 + 119790888b1) * q^87 + (1535628b3 + 1535628b2 - 324643746b1 + 324643746) q^89 + (-8280832b3 - 8280832b2 + 1179541384b1 - 1179541384) q^93 + (-1049202b2 - 102914436b1) * q^95 + (1516194b3 - 1390646810) q^97 + (-7531944b3 - 967470288) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 224 q^{3} + 1596 q^{5} - 22210 q^{9} + 50592 q^{11} + 369544 q^{13} - 1014288 q^{15} - 152124 q^{17} - 605024 q^{19} + 2503608 q^{23} - 322886 q^{25} + 12503680 q^{27} - 5454984 q^{29} - 6655040 q^{31}+ \cdots - 3869881152 q^{99}+O(q^{100})$ 4 * q - 224 * q^3 + 1596 * q^5 - 22210 * q^9 + 50592 * q^11 + 369544 * q^13 - 1014288 * q^15 - 152124 * q^17 - 605024 * q^19 + 2503608 * q^23 - 322886 * q^25 + 12503680 * q^27 - 5454984 * q^29 - 6655040 * q^31 + 11796288 * q^33 + 6860228 * q^37 - 42039944 * q^39 - 28309512 * q^41 - 2267680 * q^43 + 91283388 * q^45 + 35486640 * q^47 + 93958608 * q^51 + 33311940 * q^53 + 191084544 * q^55 - 17505296 * q^57 + 29419488 * q^59 + 35951692 * q^61 + 339557376 * q^65 + 170286704 * q^67 - 600653088 * q^69 - 927230976 * q^71 - 100806860 * q^73 - 560276864 * q^75 + 197493824 * q^79 - 865463530 * q^81 - 102854976 * q^83 + 1755147024 * q^85 + 239581776 * q^87 + 649287492 * q^89 - 2359082768 * q^93 - 205828872 * q^95 - 5562587240 * q^97 - 3869881152 * q^99

$\beta_{1}$	$=$	$( -\nu^{3} + 1141\nu^{2} - 1141\nu + 1299600 ) / 1300740$ (-v^3 + 1141v^2 - 1141v + 1299600) / 1300740
$\beta_{2}$	$=$	$( \nu^{3} - 1141\nu^{2} + 2602621\nu - 1299600 ) / 650370$ (v^3 - 1141v^2 + 2602621v - 1299600) / 650370
$\beta_{3}$	$=$	$( 4\nu^{3} + 6842 ) / 1141$ (4*v^3 + 6842) / 1141

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