LMFDB - Newform orbit 150.14.c.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,14,Mod(49,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.49"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8192,0,-93312,0,0,-1062882,0,13224840] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$160.846393428$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 64 i q^{2} + 729 i q^{3} - 4096 q^{4} - 46656 q^{6} + 176336 i q^{7} - 262144 i q^{8} - 531441 q^{9} + 6612420 q^{11} - 2985984 i q^{12} + 24028978 i q^{13} - 11285504 q^{14} + 16777216 q^{16} - 154665054 i q^{17} + \cdots - 3514111097220 q^{99} +O(q^{100}) $ q + 64i q^2 + 729i q^3 - 4096 * q^4 - 46656 * q^6 + 176336i q^7 - 262144i q^8 - 531441 * q^9 + 6612420 * q^11 - 2985984i q^12 + 24028978i q^13 - 11285504 * q^14 + 16777216 * q^16 - 154665054i q^17 - 34012224i q^18 - 190034876 * q^19 - 128548944 * q^21 + 423194880i q^22 + 352957800i q^23 + 191102976 * q^24 - 1537854592 * q^26 - 387420489i q^27 - 722272256i q^28 + 2804086266 * q^29 + 2763661208 * q^31 + 1073741824i q^32 + 4820454180i q^33 + 9898563456 * q^34 + 2176782336 * q^36 + 20030257622i q^37 - 12162232064i q^38 - 17517124962 * q^39 - 39624547206 * q^41 - 8227132416i q^42 + 81486174844i q^43 - 27084472320 * q^44 - 22589299200 * q^46 - 34136017440i q^47 + 12230590464i q^48 + 65794625511 * q^49 + 112750824366 * q^51 - 98422693888i q^52 + 21810829986i q^53 + 24794911296 * q^54 + 46225424384 * q^56 - 138535424604i q^57 + 179461521024i q^58 - 229219661220 * q^59 + 9799736750 * q^61 + 176874317312i q^62 - 93712180176i q^63 - 68719476736 * q^64 - 308509067520 * q^66 + 789042707996i q^67 + 633508061184i q^68 - 257306236200 * q^69 - 369504705240 * q^71 + 139314069504i q^72 + 693077725078i q^73 - 1281936487808 * q^74 + 778382852096 * q^76 + 1166007693120i q^77 - 1121095997568i q^78 - 2231309995208 * q^79 + 282429536481 * q^81 - 2535971021184i q^82 - 2084328707772i q^83 + 526536474624 * q^84 - 5215115190016 * q^86 + 2044178887914i q^87 - 1733406228480i q^88 - 2221961096538 * q^89 - 4237173864608 * q^91 - 1445715148800i q^92 + 2014709020632i q^93 + 2184705116160 * q^94 - 782757789696 * q^96 + 10268379896642i q^97 + 4210856032704i q^98 - 3514111097220 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8192 q^{4} - 93312 q^{6} - 1062882 q^{9} + 13224840 q^{11} - 22571008 q^{14} + 33554432 q^{16} - 380069752 q^{19} - 257097888 q^{21} + 382205952 q^{24} - 3075709184 q^{26} + 5608172532 q^{29} + 5527322416 q^{31}+ \cdots - 7028222194440 q^{99}+O(q^{100}) $ 2 * q - 8192 * q^4 - 93312 * q^6 - 1062882 * q^9 + 13224840 * q^11 - 22571008 * q^14 + 33554432 * q^16 - 380069752 * q^19 - 257097888 * q^21 + 382205952 * q^24 - 3075709184 * q^26 + 5608172532 * q^29 + 5527322416 * q^31 + 19797126912 * q^34 + 4353564672 * q^36 - 35034249924 * q^39 - 79249094412 * q^41 - 54168944640 * q^44 - 45178598400 * q^46 + 131589251022 * q^49 + 225501648732 * q^51 + 49589822592 * q^54 + 92450848768 * q^56 - 458439322440 * q^59 + 19599473500 * q^61 - 137438953472 * q^64 - 617018135040 * q^66 - 514612472400 * q^69 - 739009410480 * q^71 - 2563872975616 * q^74 + 1556765704192 * q^76 - 4462619990416 * q^79 + 564859072962 * q^81 + 1053072949248 * q^84 - 10430230380032 * q^86 - 4443922193076 * q^89 - 8474347729216 * q^91 + 4369410232320 * q^94 - 1565515579392 * q^96 - 7028222194440 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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} $

49.1

	−	1.00000i
		1.00000i

−

64.0000i

−

729.000i

−4096.00

−46656.0

−

176336.i

262144.i

−531441.

49.2

64.0000i

729.000i

−4096.00

−46656.0

176336.i

−

262144.i

−531441.

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	150.14.c.d		2
5.b	even	2	1	inner	150.14.c.d		2
5.c	odd	4	1		6.14.a.a	✓	1
5.c	odd	4	1		150.14.a.b		1
15.e	even	4	1		18.14.a.a		1
20.e	even	4	1		48.14.a.e		1
40.i	odd	4	1		192.14.a.f		1
40.k	even	4	1		192.14.a.a		1
60.l	odd	4	1		144.14.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.14.a.a	✓	1	5.c	odd	4	1
18.14.a.a		1	15.e	even	4	1
48.14.a.e		1	20.e	even	4	1
144.14.a.b		1	60.l	odd	4	1
150.14.a.b		1	5.c	odd	4	1
150.14.c.d		2	1.a	even	1	1	trivial
150.14.c.d		2	5.b	even	2	1	inner
192.14.a.a		1	40.k	even	4	1
192.14.a.f		1	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{2} + 31094384896 $ T7^2 + 31094384896 acting on $S_{14}^{\mathrm{new}}(150, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4096 $ T^2 + 4096
$3$	$ T^{2} + 531441 $ T^2 + 531441
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 31094384896 $ T^2 + 31094384896
$11$	$ (T - 6612420)^{2} $ (T - 6612420)^2
$13$	$ T^{2} + 577391783724484 $ T^2 + 577391783724484
$17$	$ T^{2} + 23\!\cdots\!16 $ T^2 + 23921278928822916
$19$	$ (T + 190034876)^{2} $ (T + 190034876)^2
$23$	$ T^{2} + 12\!\cdots\!00 $ T^2 + 124579208580840000
$29$	$ (T - 2804086266)^{2} $ (T - 2804086266)^2
$31$	$ (T - 2763661208)^{2} $ (T - 2763661208)^2
$37$	$ T^{2} + 40\!\cdots\!84 $ T^2 + 401211220403689094884
$41$	$ (T + 39624547206)^{2} $ (T + 39624547206)^2
$43$	$ T^{2} + 66\!\cdots\!36 $ T^2 + 6639996690706938424336
$47$	$ T^{2} + 11\!\cdots\!00 $ T^2 + 1165267686663984153600
$53$	$ T^{2} + 47\!\cdots\!96 $ T^2 + 475712304678196760196
$59$	$ (T + 229219661220)^{2} $ (T + 229219661220)^2
$61$	$ (T - 9799736750)^{2} $ (T - 9799736750)^2
$67$	$ T^{2} + 62\!\cdots\!16 $ T^2 + 622588395041660922336016
$71$	$ (T + 369504705240)^{2} $ (T + 369504705240)^2
$73$	$ T^{2} + 48\!\cdots\!84 $ T^2 + 480356732999295750106084
$79$	$ (T + 2231309995208)^{2} $ (T + 2231309995208)^2
$83$	$ T^{2} + 43\!\cdots\!84 $ T^2 + 4344426162042495373203984
$89$	$ (T + 2221961096538)^{2} $ (T + 2221961096538)^2
$97$	$ T^{2} + 10\!\cdots\!64 $ T^2 + 105439625701761570602876164
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Elliptic curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 64 i q^{2} + 729 i q^{3} - 4096 q^{4} - 46656 q^{6} + 176336 i q^{7} - 262144 i q^{8} - 531441 q^{9} + 6612420 q^{11} - 2985984 i q^{12} + 24028978 i q^{13} - 11285504 q^{14} + 16777216 q^{16} - 154665054 i q^{17} + \cdots - 3514111097220 q^{99} +O(q^{100}) \) q + 64i q^2 + 729i q^3 - 4096 * q^4 - 46656 * q^6 + 176336i q^7 - 262144i q^8 - 531441 * q^9 + 6612420 * q^11 - 2985984i q^12 + 24028978i q^13 - 11285504 * q^14 + 16777216 * q^16 - 154665054i q^17 - 34012224i q^18 - 190034876 * q^19 - 128548944 * q^21 + 423194880i q^22 + 352957800i q^23 + 191102976 * q^24 - 1537854592 * q^26 - 387420489i q^27 - 722272256i q^28 + 2804086266 * q^29 + 2763661208 * q^31 + 1073741824i q^32 + 4820454180i q^33 + 9898563456 * q^34 + 2176782336 * q^36 + 20030257622i q^37 - 12162232064i q^38 - 17517124962 * q^39 - 39624547206 * q^41 - 8227132416i q^42 + 81486174844i q^43 - 27084472320 * q^44 - 22589299200 * q^46 - 34136017440i q^47 + 12230590464i q^48 + 65794625511 * q^49 + 112750824366 * q^51 - 98422693888i q^52 + 21810829986i q^53 + 24794911296 * q^54 + 46225424384 * q^56 - 138535424604i q^57 + 179461521024i q^58 - 229219661220 * q^59 + 9799736750 * q^61 + 176874317312i q^62 - 93712180176i q^63 - 68719476736 * q^64 - 308509067520 * q^66 + 789042707996i q^67 + 633508061184i q^68 - 257306236200 * q^69 - 369504705240 * q^71 + 139314069504i q^72 + 693077725078i q^73 - 1281936487808 * q^74 + 778382852096 * q^76 + 1166007693120i q^77 - 1121095997568i q^78 - 2231309995208 * q^79 + 282429536481 * q^81 - 2535971021184i q^82 - 2084328707772i q^83 + 526536474624 * q^84 - 5215115190016 * q^86 + 2044178887914i q^87 - 1733406228480i q^88 - 2221961096538 * q^89 - 4237173864608 * q^91 - 1445715148800i q^92 + 2014709020632i q^93 + 2184705116160 * q^94 - 782757789696 * q^96 + 10268379896642i q^97 + 4210856032704i q^98 - 3514111097220 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8192 q^{4} - 93312 q^{6} - 1062882 q^{9} + 13224840 q^{11} - 22571008 q^{14} + 33554432 q^{16} - 380069752 q^{19} - 257097888 q^{21} + 382205952 q^{24} - 3075709184 q^{26} + 5608172532 q^{29} + 5527322416 q^{31}+ \cdots - 7028222194440 q^{99}+O(q^{100}) \) 2 * q - 8192 * q^4 - 93312 * q^6 - 1062882 * q^9 + 13224840 * q^11 - 22571008 * q^14 + 33554432 * q^16 - 380069752 * q^19 - 257097888 * q^21 + 382205952 * q^24 - 3075709184 * q^26 + 5608172532 * q^29 + 5527322416 * q^31 + 19797126912 * q^34 + 4353564672 * q^36 - 35034249924 * q^39 - 79249094412 * q^41 - 54168944640 * q^44 - 45178598400 * q^46 + 131589251022 * q^49 + 225501648732 * q^51 + 49589822592 * q^54 + 92450848768 * q^56 - 458439322440 * q^59 + 19599473500 * q^61 - 137438953472 * q^64 - 617018135040 * q^66 - 514612472400 * q^69 - 739009410480 * q^71 - 2563872975616 * q^74 + 1556765704192 * q^76 - 4462619990416 * q^79 + 564859072962 * q^81 + 1053072949248 * q^84 - 10430230380032 * q^86 - 4443922193076 * q^89 - 8474347729216 * q^91 + 4369410232320 * q^94 - 1565515579392 * q^96 - 7028222194440 * q^99

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