LMFDB - Newform orbit 288.9.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,9,Mod(271,288)]

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([1, 1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("288.271");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	288.b (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:	$117.325039698$
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} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 $ x^6 - x^5 + 78x^4 - 514x^3 + 4237x^2 - 18333x + 238980
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{33} $
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_{2} q^{5} + ( - \beta_{4} + \beta_{2}) q^{7}+O(q^{10}) $ q - b2 * q^5 + (-b4 + b2) * q^7	$ q - \beta_{2} q^{5} + ( - \beta_{4} + \beta_{2}) q^{7} + (2 \beta_{3} - 37 \beta_1 + 7824) q^{11} + (\beta_{5} + 6 \beta_{4} - 11 \beta_{2}) q^{13} + ( - 17 \beta_{3} - 94 \beta_1 + 33507) q^{17} + (78 \beta_{3} + 101 \beta_1 + 15904) q^{19} + ( - 10 \beta_{5} + \cdots - 311 \beta_{2}) q^{23}+ \cdots + (5409 \beta_{3} + 578622 \beta_1 + 20867405) q^{97}+O(q^{100}) $ q - b2 * q^5 + (-b4 + b2) * q^7 + (2b3 - 37b1 + 7824) * q^11 + (b5 + 6b4 - 11b2) * q^13 + (-17b3 - 94b1 + 33507) * q^17 + (78b3 + 101b1 + 15904) * q^19 + (-10b5 - b4 - 311b2) * q^23 + (66b3 + 2300b1 - 74633) * q^25 + (-8b5 - 176b4 + 181b2) q^29 + (-2b5 - 64b4 - 1432b2) q^31 + (588b3 - 280b1 + 331716) * q^35 + (-49b5 + 218b4 + 339b2) q^37 + (730b3 - 27188b1 - 635940) * q^41 + (-84b3 - 7287b1 + 1646890) * q^43 + (86b5 - 862b4 + 1190b2) q^47 + (-1848b3 - 67984b1 - 2109863) * q^49 + (-49b5 - 1702b4 + 4725b2) q^53 + (152b5 + 739b4 - 8387b2) q^55 + (1856b3 - 25249b1 + 4207866) * q^59 + (425b5 + 630b4 + 137b2) q^61 + (-9054b3 + 166460b1 - 4513098) * q^65 + (5022b3 + 54021b1 - 7973360) * q^67 + (-358b5 + 4229b4 + 22899b2) q^71 + (-981b3 + 184234b1 - 9247813) * q^73 + (-35b5 - 6482b4 + 18746b2) q^77 + (392b5 + 14494b4 + 3778b2) q^79 + (12464b3 + 133599b1 + 24111882) * q^83 + (-475b5 - 6690b4 - 47104b2) q^85 + (1981b3 + 415366b1 - 23694915) * q^89 + (-3444b3 - 142744b1 + 45660804) * q^91 + (2840b5 + 30365b4 + 31619b2) q^95 + (5409b3 + 578622b1 + 20867405) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 46940 q^{11} + 201076 q^{17} + 95268 q^{19} - 447930 q^{25} + 1989120 q^{35} - 3817100 q^{41} + 9881508 q^{43} - 12655482 q^{49} + 25243484 q^{59} - 27060480 q^{65} - 47850204 q^{67} - 55484916 q^{73} + 144646364 q^{83} - 142173452 q^{89} + 273971712 q^{91} + 125193612 q^{97}+O(q^{100}) $ 6 * q + 46940 * q^11 + 201076 * q^17 + 95268 * q^19 - 447930 * q^25 + 1989120 * q^35 - 3817100 * q^41 + 9881508 * q^43 - 12655482 * q^49 + 25243484 * q^59 - 27060480 * q^65 - 47850204 * q^67 - 55484916 * q^73 + 144646364 * q^83 - 142173452 * q^89 + 273971712 * q^91 + 125193612 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 $ x^6 - x^5 + 78*x^4 - 514*x^3 + 4237*x^2 - 18333*x + 238980 :

$\beta_{1}$	$=$	$ ( \nu^{5} - 34\nu^{4} + 176\nu^{3} - 1202\nu^{2} + 12159\nu - 72444 ) / 1024 $ (v^5 - 34v^4 + 176v^3 - 1202v^2 + 12159v - 72444) / 1024
$\beta_{2}$	$=$	$ ( -7\nu^{5} + 46\nu^{4} + 368\nu^{3} + 4766\nu^{2} - 5305\nu - 46620 ) / 512 $ (-7v^5 + 46v^4 + 368v^3 + 4766v^2 - 5305*v - 46620) / 512
$\beta_{3}$	$=$	$ ( 29\nu^{5} + 38\nu^{4} + 2032\nu^{3} - 15402\nu^{2} - 12957\nu - 349324 ) / 512 $ (29v^5 + 38v^4 + 2032v^3 - 15402v^2 - 12957*v - 349324) / 512
$\beta_{4}$	$=$	$ ( 31\nu^{5} + 162\nu^{4} + 784\nu^{3} + 18610\nu^{2} - 35679\nu + 900732 ) / 512 $ (31v^5 + 162v^4 + 784v^3 + 18610v^2 - 35679*v + 900732) / 512
$\beta_{5}$	$=$	$ ( 19\nu^{5} - 6\nu^{4} + 1424\nu^{3} - 6582\nu^{2} + 137709\nu - 198324 ) / 32 $ (19v^5 - 6v^4 + 1424v^3 - 6582v^2 + 137709*v - 198324) / 32

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

$\nu$	$=$	$ ( \beta_{5} - 2\beta_{4} - 8\beta_{3} - 2\beta_{2} - 48\beta _1 + 680 ) / 4096 $ (b5 - 2b4 - 8b3 - 2b2 - 48b1 + 680) / 4096
$\nu^{2}$	$=$	$ ( -\beta_{5} + 66\beta_{4} - 56\beta_{3} + 66\beta_{2} + 688\beta _1 - 105832 ) / 4096 $ (-b5 + 66b4 - 56b3 + 66b2 + 688b1 - 105832) / 4096
$\nu^{3}$	$=$	$ ( -11\beta_{5} + 22\beta_{4} + 472\beta_{3} + 2070\beta_{2} + 6928\beta _1 + 893768 ) / 4096 $ (-11b5 + 22b4 + 472b3 + 2070b2 + 6928*b1 + 893768) / 4096
$\nu^{4}$	$=$	$ ( 343\beta_{5} - 1902\beta_{4} + 1672\beta_{3} + 4242\beta_{2} - 128208\beta _1 - 2071336 ) / 4096 $ (343b5 - 1902b4 + 1672b3 + 4242b2 - 128208*b1 - 2071336) / 4096
$\nu^{5}$	$=$	$ ( 237\beta_{5} + 35110\beta_{4} + 3736\beta_{3} - 116442\beta_{2} + 26512\beta _1 - 66476152 ) / 4096 $ (237b5 + 35110b4 + 3736b3 - 116442b2 + 26512*b1 - 66476152) / 4096

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} $

271.1

−0.793226	−	7.99733i
5.83239	+	4.16154i
−4.53916	+	7.17456i
−4.53916	−	7.17456i
5.83239	−	4.16154i
−0.793226	+	7.99733i

−

816.841i

1333.52i

271.2

−

796.785i

−

1779.55i

271.3

−

306.178i

4321.70i

271.4

306.178i

−

4321.70i

271.5

796.785i

1779.55i

271.6

816.841i

−

1333.52i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.9.b.b		6
3.b	odd	2	1		32.9.d.b		6
4.b	odd	2	1		72.9.b.b		6
8.b	even	2	1		72.9.b.b		6
8.d	odd	2	1	inner	288.9.b.b		6
12.b	even	2	1		8.9.d.b	✓	6
24.f	even	2	1		32.9.d.b		6
24.h	odd	2	1		8.9.d.b	✓	6
48.i	odd	4	2		256.9.c.n		12
48.k	even	4	2		256.9.c.n		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.9.d.b	✓	6	12.b	even	2	1
8.9.d.b	✓	6	24.h	odd	2	1
32.9.d.b		6	3.b	odd	2	1
32.9.d.b		6	24.f	even	2	1
72.9.b.b		6	4.b	odd	2	1
72.9.b.b		6	8.b	even	2	1
256.9.c.n		12	48.i	odd	4	2
256.9.c.n		12	48.k	even	4	2
288.9.b.b		6	1.a	even	1	1	trivial
288.9.b.b		6	8.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 1395840T_{5}^{4} + 545665781760T_{5}^{2} + 39710399791104000 $ T5^6 + 1395840*T5^4 + 545665781760*T5^2 + 39710399791104000 acting on $S_{9}^{\mathrm{new}}(288, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + \cdots + 39\!\cdots\!00 $ T^6 + 1395840T^4 + 545665781760T^2 + 39710399791104000
$7$	$ T^{6} + \cdots + 10\!\cdots\!40 $ T^6 + 23622144T^4 + 97991030538240T^2 + 105178925880091607040
$11$	$ (T^{3} - 23470 T^{2} + \cdots - 155265768392)^{2} $ (T^3 - 23470T^2 + 138543932T - 155265768392)^2
$13$	$ T^{6} + \cdots + 54\!\cdots\!40 $ T^6 + 2916883584T^4 + 1906173291186032640T^2 + 54882278056166576215818240
$17$	$ (T^{3} + \cdots + 13136750437640)^{2} $ (T^3 - 100538T^2 + 1302520076T + 13136750437640)^2
$19$	$ (T^{3} + \cdots + 34\!\cdots\!08)^{2} $ (T^3 - 47634T^2 - 40091765700T + 3473468180164808)^2
$23$	$ T^{6} + \cdots + 10\!\cdots\!00 $ T^6 + 339714875904T^4 + 24596517798037384396800T^2 + 106705426531333746655362023424000
$29$	$ T^{6} + \cdots + 21\!\cdots\!00 $ T^6 + 839474501760T^4 + 231826558755136128552960T^2 + 21114648968930067499784837922816000
$31$	$ T^{6} + \cdots + 10\!\cdots\!00 $ T^6 + 3032572108800T^4 + 2257947076541143842816000T^2 + 107010007724344350329471080857600000
$37$	$ T^{6} + \cdots + 11\!\cdots\!40 $ T^6 + 6554265773184T^4 + 11058362162648417007636480T^2 + 1153788351697704319628784876666224640
$41$	$ (T^{3} + \cdots - 26\!\cdots\!28)^{2} $ (T^3 + 1908550T^2 - 12427883610868T - 26999644127823495928)^2
$43$	$ (T^{3} + \cdots - 30\!\cdots\!00)^{2} $ (T^3 - 4940754T^2 + 7347988694844T - 3041301161027731000)^2
$47$	$ T^{6} + \cdots + 35\!\cdots\!40 $ T^6 + 35076807051264T^4 + 79981661029486572805816320T^2 + 35509683309207254778351568007347568640
$53$	$ T^{6} + \cdots + 66\!\cdots\!40 $ T^6 + 94971255986304T^4 + 2362516759437555884997611520T^2 + 6601592212144193096692191217614811299840
$59$	$ (T^{3} + \cdots + 70\!\cdots\!48)^{2} $ (T^3 - 12621742T^2 + 21654746091836T + 70683721047699464248)^2
$61$	$ T^{6} + \cdots + 12\!\cdots\!00 $ T^6 + 379841020076160T^4 + 42641906955683759757513461760T^2 + 1267629533502933649024864886336523730944000
$67$	$ (T^{3} + \cdots - 12\!\cdots\!60)^{2} $ (T^3 + 23925102T^2 - 20484765801924T - 1277567970410307279160)^2
$71$	$ T^{6} + \cdots + 10\!\cdots\!00 $ T^6 + 1529204547064320T^4 + 719459944649095340794733199360T^2 + 106311015001033381244601086306980699570176000
$73$	$ (T^{3} + \cdots - 13\!\cdots\!20)^{2} $ (T^3 + 27742458T^2 - 219164368244724T - 1342953361145769894920)^2
$79$	$ T^{6} + \cdots + 16\!\cdots\!00 $ T^6 + 5102910846289920T^4 + 7181508231688373010663509852160T^2 + 1674634634346989364258402890095174757646336000
$83$	$ (T^{3} + \cdots + 15\!\cdots\!40)^{2} $ (T^3 - 72323182T^2 + 443882973572156T + 15873727783266586237240)^2
$89$	$ (T^{3} + \cdots - 46\!\cdots\!72)^{2} $ (T^3 + 71086726T^2 - 742202618348020T - 46893283645168391278072)^2
$97$	$ (T^{3} + \cdots + 47\!\cdots\!60)^{2} $ (T^3 - 62596806T^2 - 3560740321943796T + 47468642276606493351160)^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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	288.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{5} + ( - \beta_{4} + \beta_{2}) q^{7}+O(q^{10}) \) q - b2 * q^5 + (-b4 + b2) * q^7	\( q - \beta_{2} q^{5} + ( - \beta_{4} + \beta_{2}) q^{7} + (2 \beta_{3} - 37 \beta_1 + 7824) q^{11} + (\beta_{5} + 6 \beta_{4} - 11 \beta_{2}) q^{13} + ( - 17 \beta_{3} - 94 \beta_1 + 33507) q^{17} + (78 \beta_{3} + 101 \beta_1 + 15904) q^{19} + ( - 10 \beta_{5} + \cdots - 311 \beta_{2}) q^{23}+ \cdots + (5409 \beta_{3} + 578622 \beta_1 + 20867405) q^{97}+O(q^{100}) \) q - b2 * q^5 + (-b4 + b2) * q^7 + (2b3 - 37b1 + 7824) * q^11 + (b5 + 6b4 - 11b2) * q^13 + (-17b3 - 94b1 + 33507) * q^17 + (78b3 + 101b1 + 15904) * q^19 + (-10b5 - b4 - 311b2) * q^23 + (66b3 + 2300b1 - 74633) * q^25 + (-8b5 - 176b4 + 181b2) q^29 + (-2b5 - 64b4 - 1432b2) q^31 + (588b3 - 280b1 + 331716) * q^35 + (-49b5 + 218b4 + 339b2) q^37 + (730b3 - 27188b1 - 635940) * q^41 + (-84b3 - 7287b1 + 1646890) * q^43 + (86b5 - 862b4 + 1190b2) q^47 + (-1848b3 - 67984b1 - 2109863) * q^49 + (-49b5 - 1702b4 + 4725b2) q^53 + (152b5 + 739b4 - 8387b2) q^55 + (1856b3 - 25249b1 + 4207866) * q^59 + (425b5 + 630b4 + 137b2) q^61 + (-9054b3 + 166460b1 - 4513098) * q^65 + (5022b3 + 54021b1 - 7973360) * q^67 + (-358b5 + 4229b4 + 22899b2) q^71 + (-981b3 + 184234b1 - 9247813) * q^73 + (-35b5 - 6482b4 + 18746b2) q^77 + (392b5 + 14494b4 + 3778b2) q^79 + (12464b3 + 133599b1 + 24111882) * q^83 + (-475b5 - 6690b4 - 47104b2) q^85 + (1981b3 + 415366b1 - 23694915) * q^89 + (-3444b3 - 142744b1 + 45660804) * q^91 + (2840b5 + 30365b4 + 31619b2) q^95 + (5409b3 + 578622b1 + 20867405) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 46940 q^{11} + 201076 q^{17} + 95268 q^{19} - 447930 q^{25} + 1989120 q^{35} - 3817100 q^{41} + 9881508 q^{43} - 12655482 q^{49} + 25243484 q^{59} - 27060480 q^{65} - 47850204 q^{67} - 55484916 q^{73} + 144646364 q^{83} - 142173452 q^{89} + 273971712 q^{91} + 125193612 q^{97}+O(q^{100}) \) 6 * q + 46940 * q^11 + 201076 * q^17 + 95268 * q^19 - 447930 * q^25 + 1989120 * q^35 - 3817100 * q^41 + 9881508 * q^43 - 12655482 * q^49 + 25243484 * q^59 - 27060480 * q^65 - 47850204 * q^67 - 55484916 * q^73 + 144646364 * q^83 - 142173452 * q^89 + 273971712 * q^91 + 125193612 * q^97

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