LMFDB - Newform orbit 288.8.d.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(145,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("288.145");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.d (of order $2$, degree $1$, 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:	$89.9668873394$
Analytic rank:	$0$
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} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 $ x^6 - 3x^5 - 10x^4 - 24x^3 - 320x^2 - 3072*x + 32768
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{3} - \beta_{2}) q^{5} + (\beta_1 + 115) q^{7}+O(q^{10}) $ q + (-b3 - b2) * q^5 + (b1 + 115) * q^7	$ q + ( - \beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 115) q^{7} + (\beta_{5} + 7 \beta_{3} + 3 \beta_{2}) q^{11} + ( - 2 \beta_{5} - 3 \beta_{3} - 21 \beta_{2}) q^{13} + (7 \beta_{4} - 25 \beta_1 - 248) q^{17} + ( - 5 \beta_{5} + 93 \beta_{3} + 105 \beta_{2}) q^{19} + (8 \beta_{4} - 11 \beta_1 - 217) q^{23} + ( - 6 \beta_{4} - 38 \beta_1 - 6567) q^{25} + ( - 16 \beta_{5} + \cdots + 2337 \beta_{2}) q^{29}+ \cdots + (849 \beta_{4} - 7183 \beta_1 - 183496) q^{97}+O(q^{100}) $ q + (-b3 - b2) * q^5 + (b1 + 115) * q^7 + (b5 + 7b3 + 3b2) * q^11 + (-2b5 - 3b3 - 21b2) q^13 + (7b4 - 25b1 - 248) * q^17 + (-5b5 + 93b3 + 105b2) q^19 + (8b4 - 11b1 - 217) * q^23 + (-6b4 - 38b1 - 6567) * q^25 + (-16b5 - 79b3 + 2337b2) q^29 + (72b4 + 12b1 + 14908) * q^31 + (-50b5 - 478b3 - 368b2) q^35 + (58b5 - 33b3 + 6633b2) q^37 + (-142b4 - 302b1 - 87022) * q^41 + (38b5 + 522b3 - 579b2) q^43 + (168b4 + 210b1 + 261198) * q^47 + (-168b4 - 168b1 - 85287) * q^49 + (170b5 - 535b3 - 17949b2) q^53 + (-96b4 + 1237b1 + 545423) * q^55 + (224b5 - 3680b3 - 10933b2) q^59 + (-66b5 + 3993b3 - 7353b2) q^61 + (50b4 - 2030b1 - 236740) * q^65 + (123b5 - 1443b3 + 4017b2) q^67 + (504b4 + 3279b1 - 1264923) * q^71 + (1239b4 - 809b1 + 348404) * q^73 + (-218b5 + 12598b3 - 31860b2) q^77 + (2016b4 - 2054b1 - 2669330) * q^79 + (-568b5 - 9992b3 + 13319b2) q^83 + (1390b5 + 9876b3 - 85134b2) q^85 + (-439b4 - 1079b1 - 362020) * q^89 + (1142b5 - 19782b3 + 56016b2) q^91 + (1184b4 - 1313b1 + 8089613) * q^95 + (849b4 - 7183b1 - 183496) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 688 q^{7}+O(q^{10}) $ 6 * q + 688 * q^7	$ 6 q + 688 q^{7} - 1452 q^{17} - 1296 q^{23} - 39314 q^{25} + 89280 q^{31} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} - 1416480 q^{65} - 7597104 q^{71} + 2089564 q^{73} - 16015904 q^{79} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) $ 6 * q + 688 * q^7 - 1452 * q^17 - 1296 * q^23 - 39314 * q^25 + 89280 * q^31 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 - 1416480 * q^65 - 7597104 * q^71 + 2089564 * q^73 - 16015904 * q^79 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 $ x^6 - 3*x^5 - 10*x^4 - 24*x^3 - 320*x^2 - 3072*x + 32768 :

$\beta_{1}$	$=$	$ ( -\nu^{4} + 3\nu^{3} + 42\nu^{2} - 72\nu + 106 ) / 2 $ (-v^4 + 3v^3 + 42v^2 - 72*v + 106) / 2
$\beta_{2}$	$=$	$ ( -3\nu^{5} - 7\nu^{4} - 50\nu^{3} + 104\nu^{2} + 1088\nu + 15872 ) / 256 $ (-3v^5 - 7v^4 - 50v^3 + 104v^2 + 1088*v + 15872) / 256
$\beta_{3}$	$=$	$ ( -11\nu^{5} - 47\nu^{4} + 222\nu^{3} - 1112\nu^{2} + 3136\nu + 60928 ) / 256 $ (-11v^5 - 47v^4 + 222v^3 - 1112v^2 + 3136*v + 60928) / 256
$\beta_{4}$	$=$	$ ( -\nu^{5} + \nu^{4} + 16\nu^{3} + 108\nu^{2} + 1200\nu + 2260 ) / 4 $ (-v^5 + v^4 + 16v^3 + 108v^2 + 1200*v + 2260) / 4
$\beta_{5}$	$=$	$ ( 93\nu^{5} - 487\nu^{4} - 1458\nu^{3} - 3352\nu^{2} + 101952\nu - 205312 ) / 256 $ (93v^5 - 487v^4 - 1458v^3 - 3352v^2 + 101952*v - 205312) / 256

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

$\nu$	$=$	$ ( \beta_{5} + 2\beta_{4} - \beta_{3} - 8\beta_{2} - 2\beta _1 + 512 ) / 1024 $ (b5 + 2b4 - b3 - 8b2 - 2*b1 + 512) / 1024
$\nu^{2}$	$=$	$ ( -\beta_{5} + 6\beta_{4} - 63\beta_{3} + 72\beta_{2} + 26\beta _1 + 4960 ) / 1024 $ (-b5 + 6b4 - 63b3 + 72b2 + 26b1 + 4960) / 1024
$\nu^{3}$	$=$	$ ( -9\beta_{5} + 38\beta_{4} + 329\beta_{3} - 2296\beta_{2} + 58\beta _1 + 32288 ) / 1024 $ (-9b5 + 38b4 + 329b3 - 2296b2 + 58*b1 + 32288) / 1024
$\nu^{4}$	$=$	$ ( -141\beta_{5} + 222\beta_{4} - 1587\beta_{3} - 3288\beta_{2} - 638\beta _1 + 376864 ) / 1024 $ (-141b5 + 222b4 - 1587b3 - 3288b2 - 638*b1 + 376864) / 1024
$\nu^{5}$	$=$	$ ( 807\beta_{5} - 218\beta_{4} - 4327\beta_{3} - 41848\beta_{2} + 698\beta _1 + 4357792 ) / 1024 $ (807b5 - 218b4 - 4327b3 - 41848b2 + 698*b1 + 4357792) / 1024

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$1$

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} $

145.1

5.57668	−	0.949035i
−4.85268	+	2.90715i
0.776001	−	5.60338i
0.776001	+	5.60338i
−4.85268	−	2.90715i
5.57668	+	0.949035i

−

338.443i

438.996

145.2

−

324.492i

956.960

145.3

−

184.916i

−1051.96

145.4

184.916i

−1051.96

145.5

324.492i

956.960

145.6

338.443i

438.996

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.8.d.b		6
3.b	odd	2	1		32.8.b.a		6
4.b	odd	2	1		72.8.d.b		6
8.b	even	2	1	inner	288.8.d.b		6
8.d	odd	2	1		72.8.d.b		6
12.b	even	2	1		8.8.b.a	✓	6
24.f	even	2	1		8.8.b.a	✓	6
24.h	odd	2	1		32.8.b.a		6
48.i	odd	4	2		256.8.a.q		6
48.k	even	4	2		256.8.a.r		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.8.b.a	✓	6	12.b	even	2	1
8.8.b.a	✓	6	24.f	even	2	1
32.8.b.a		6	3.b	odd	2	1
32.8.b.a		6	24.h	odd	2	1
72.8.d.b		6	4.b	odd	2	1
72.8.d.b		6	8.d	odd	2	1
256.8.a.q		6	48.i	odd	4	2
256.8.a.r		6	48.k	even	4	2
288.8.d.b		6	1.a	even	1	1	trivial
288.8.d.b		6	8.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 254032T_{5}^{4} + 19577926400T_{5}^{2} + 412405245440000 $ T5^6 + 254032*T5^4 + 19577926400*T5^2 + 412405245440000 acting on $S_{8}^{\mathrm{new}}(288, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + \cdots + 412405245440000 $ T^6 + 254032T^4 + 19577926400T^2 + 412405245440000
$7$	$ (T^{3} - 344 T^{2} + \cdots + 441929216)^{2} $ (T^3 - 344T^2 - 1048384T + 441929216)^2
$11$	$ T^{6} + \cdots + 21\!\cdots\!00 $ T^6 + 52294004T^4 + 743173813625904T^2 + 2149855992230386161600
$13$	$ T^{6} + \cdots + 24\!\cdots\!00 $ T^6 + 171080144T^4 + 5821641670624000T^2 + 24487251428999370240000
$17$	$ (T^{3} + \cdots + 9112197964104)^{2} $ (T^3 + 726T^2 - 907714996T + 9112197964104)^2
$19$	$ T^{6} + \cdots + 47\!\cdots\!04 $ T^6 + 3360814100T^4 + 2904356007201913648T^2 + 475867300669609431337619904
$23$	$ (T^{3} + \cdots + 2134822184448)^{2} $ (T^3 + 648T^2 - 505954624T + 2134822184448)^2
$29$	$ T^{6} + \cdots + 42\!\cdots\!00 $ T^6 + 55662621776T^4 + 786969033263564256000T^2 + 424205494617184221641909760000
$31$	$ (T^{3} + \cdots - 18\!\cdots\!28)^{2} $ (T^3 - 44640T^2 - 34183492608T - 1867811583459328)^2
$37$	$ T^{6} + \cdots + 61\!\cdots\!64 $ T^6 + 490654094672T^4 + 62693545808917922763520T^2 + 617992928421224147868032643723264
$41$	$ (T^{3} + \cdots - 17\!\cdots\!00)^{2} $ (T^3 + 260622T^2 - 233067695380T - 17679961159949400)^2
$43$	$ T^{6} + \cdots + 77\!\cdots\!56 $ T^6 + 124911737588T^4 + 1911474156456083249200T^2 + 7785148142341185674388965355456
$47$	$ (T^{3} + \cdots + 15\!\cdots\!96)^{2} $ (T^3 - 783216T^2 - 49048909056T + 15251091035959296)^2
$53$	$ T^{6} + \cdots + 12\!\cdots\!36 $ T^6 + 3916631783120T^4 + 4181846884553965477356288T^2 + 1206327310526039972497627084133953536
$59$	$ T^{6} + \cdots + 55\!\cdots\!04 $ T^6 + 6619585104052T^4 + 11119713619132481564744240T^2 + 550737065763712802555226997336378304
$61$	$ T^{6} + \cdots + 10\!\cdots\!00 $ T^6 + 4505952081744T^4 + 5487833931913656286089984T^2 + 1053139851314178567056917443970560000
$67$	$ T^{6} + \cdots + 75\!\cdots\!24 $ T^6 + 1291377394260T^4 + 126799666134501803239728T^2 + 756734382489347558070672940185024
$71$	$ (T^{3} + \cdots + 38\!\cdots\!92)^{2} $ (T^3 + 3798552T^2 - 9464693891904T + 3841164241580155392)^2
$73$	$ (T^{3} + \cdots + 21\!\cdots\!72)^{2} $ (T^3 - 1044782T^2 - 9934777446036T + 216053954068698072)^2
$79$	$ (T^{3} + \cdots - 49\!\cdots\!40)^{2} $ (T^3 + 8007952T^2 - 7783396574464T - 49251461081810391040)^2
$83$	$ T^{6} + \cdots + 63\!\cdots\!16 $ T^6 + 37884069033748T^4 + 110669295989097617184580400T^2 + 63652853266017468506124209929689530816
$89$	$ (T^{3} + \cdots - 11\!\cdots\!20)^{2} $ (T^3 + 1084542T^2 - 2404948069204T - 1174636932800532120)^2
$97$	$ (T^{3} + \cdots - 16\!\cdots\!24)^{2} $ (T^3 + 544154T^2 - 57460066941556T - 1662041066006495624)^2
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 115) q^{7}+O(q^{10}) \) q + (-b3 - b2) * q^5 + (b1 + 115) * q^7	\( q + ( - \beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 115) q^{7} + (\beta_{5} + 7 \beta_{3} + 3 \beta_{2}) q^{11} + ( - 2 \beta_{5} - 3 \beta_{3} - 21 \beta_{2}) q^{13} + (7 \beta_{4} - 25 \beta_1 - 248) q^{17} + ( - 5 \beta_{5} + 93 \beta_{3} + 105 \beta_{2}) q^{19} + (8 \beta_{4} - 11 \beta_1 - 217) q^{23} + ( - 6 \beta_{4} - 38 \beta_1 - 6567) q^{25} + ( - 16 \beta_{5} + \cdots + 2337 \beta_{2}) q^{29}+ \cdots + (849 \beta_{4} - 7183 \beta_1 - 183496) q^{97}+O(q^{100}) \) q + (-b3 - b2) * q^5 + (b1 + 115) * q^7 + (b5 + 7b3 + 3b2) * q^11 + (-2b5 - 3b3 - 21b2) q^13 + (7b4 - 25b1 - 248) * q^17 + (-5b5 + 93b3 + 105b2) q^19 + (8b4 - 11b1 - 217) * q^23 + (-6b4 - 38b1 - 6567) * q^25 + (-16b5 - 79b3 + 2337b2) q^29 + (72b4 + 12b1 + 14908) * q^31 + (-50b5 - 478b3 - 368b2) q^35 + (58b5 - 33b3 + 6633b2) q^37 + (-142b4 - 302b1 - 87022) * q^41 + (38b5 + 522b3 - 579b2) q^43 + (168b4 + 210b1 + 261198) * q^47 + (-168b4 - 168b1 - 85287) * q^49 + (170b5 - 535b3 - 17949b2) q^53 + (-96b4 + 1237b1 + 545423) * q^55 + (224b5 - 3680b3 - 10933b2) q^59 + (-66b5 + 3993b3 - 7353b2) q^61 + (50b4 - 2030b1 - 236740) * q^65 + (123b5 - 1443b3 + 4017b2) q^67 + (504b4 + 3279b1 - 1264923) * q^71 + (1239b4 - 809b1 + 348404) * q^73 + (-218b5 + 12598b3 - 31860b2) q^77 + (2016b4 - 2054b1 - 2669330) * q^79 + (-568b5 - 9992b3 + 13319b2) q^83 + (1390b5 + 9876b3 - 85134b2) q^85 + (-439b4 - 1079b1 - 362020) * q^89 + (1142b5 - 19782b3 + 56016b2) q^91 + (1184b4 - 1313b1 + 8089613) * q^95 + (849b4 - 7183b1 - 183496) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 688 q^{7}+O(q^{10}) \) 6 * q + 688 * q^7	\( 6 q + 688 q^{7} - 1452 q^{17} - 1296 q^{23} - 39314 q^{25} + 89280 q^{31} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} - 1416480 q^{65} - 7597104 q^{71} + 2089564 q^{73} - 16015904 q^{79} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) 6 * q + 688 * q^7 - 1452 * q^17 - 1296 * q^23 - 39314 * q^25 + 89280 * q^31 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 - 1416480 * q^65 - 7597104 * q^71 + 2089564 * q^73 - 16015904 * q^79 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

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