LMFDB - Newform orbit 150.8.c.j

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,8,Mod(49,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-128,0,432,0,0,-1458,0,13104] 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:	$46.8577538226$
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 30)
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 + 8 i q^{2} - 27 i q^{3} - 64 q^{4} + 216 q^{6} + 1084 i q^{7} - 512 i q^{8} - 729 q^{9} + 6552 q^{11} + 1728 i q^{12} + 2522 i q^{13} - 8672 q^{14} + 4096 q^{16} - 30486 i q^{17} - 5832 i q^{18} - 12020 q^{19} + \cdots - 4776408 q^{99} +O(q^{100}) $ q + 8i q^2 - 27i q^3 - 64 * q^4 + 216 * q^6 + 1084i q^7 - 512i q^8 - 729 * q^9 + 6552 * q^11 + 1728i q^12 + 2522i q^13 - 8672 * q^14 + 4096 * q^16 - 30486i q^17 - 5832i q^18 - 12020 * q^19 + 29268 * q^21 + 52416i q^22 - 3528i q^23 - 13824 * q^24 - 20176 * q^26 + 19683i q^27 - 69376i q^28 - 84930 * q^29 - 94168 * q^31 + 32768i q^32 - 176904i q^33 + 243888 * q^34 + 46656 * q^36 + 509974i q^37 - 96160i q^38 + 68094 * q^39 + 841002 * q^41 + 234144i q^42 + 889052i q^43 - 419328 * q^44 + 28224 * q^46 + 852264i q^47 - 110592i q^48 - 351513 * q^49 - 823122 * q^51 - 161408i q^52 + 1594482i q^53 - 157464 * q^54 + 555008 * q^56 + 324540i q^57 - 679440i q^58 - 752040 * q^59 + 1538702 * q^61 - 753344i q^62 - 790236i q^63 - 262144 * q^64 + 1415232 * q^66 + 947524i q^67 + 1951104i q^68 - 95256 * q^69 - 3824928 * q^71 + 373248i q^72 - 913678i q^73 - 4079792 * q^74 + 769280 * q^76 + 7102368i q^77 + 544752i q^78 - 1621880 * q^79 + 531441 * q^81 + 6728016i q^82 + 1470012i q^83 - 1873152 * q^84 - 7112416 * q^86 + 2293110i q^87 - 3354624i q^88 - 2375010 * q^89 - 2733848 * q^91 + 225792i q^92 + 2542536i q^93 - 6818112 * q^94 + 884736 * q^96 - 14138786i q^97 - 2812104i q^98 - 4776408 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} + 432 q^{6} - 1458 q^{9} + 13104 q^{11} - 17344 q^{14} + 8192 q^{16} - 24040 q^{19} + 58536 q^{21} - 27648 q^{24} - 40352 q^{26} - 169860 q^{29} - 188336 q^{31} + 487776 q^{34} + 93312 q^{36}+ \cdots - 9552816 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 + 432 * q^6 - 1458 * q^9 + 13104 * q^11 - 17344 * q^14 + 8192 * q^16 - 24040 * q^19 + 58536 * q^21 - 27648 * q^24 - 40352 * q^26 - 169860 * q^29 - 188336 * q^31 + 487776 * q^34 + 93312 * q^36 + 136188 * q^39 + 1682004 * q^41 - 838656 * q^44 + 56448 * q^46 - 703026 * q^49 - 1646244 * q^51 - 314928 * q^54 + 1110016 * q^56 - 1504080 * q^59 + 3077404 * q^61 - 524288 * q^64 + 2830464 * q^66 - 190512 * q^69 - 7649856 * q^71 - 8159584 * q^74 + 1538560 * q^76 - 3243760 * q^79 + 1062882 * q^81 - 3746304 * q^84 - 14224832 * q^86 - 4750020 * q^89 - 5467696 * q^91 - 13636224 * q^94 + 1769472 * q^96 - 9552816 * 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

−

8.00000i

27.0000i

−64.0000

216.000

−

1084.00i

512.000i

−729.000

49.2

8.00000i

−

27.0000i

−64.0000

216.000

1084.00i

−

512.000i

−729.000

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	150.8.c.j		2
3.b	odd	2	1		450.8.c.b		2
5.b	even	2	1	inner	150.8.c.j		2
5.c	odd	4	1		30.8.a.a	✓	1
5.c	odd	4	1		150.8.a.q		1
15.d	odd	2	1		450.8.c.b		2
15.e	even	4	1		90.8.a.i		1
15.e	even	4	1		450.8.a.k		1
20.e	even	4	1		240.8.a.k		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.8.a.a	✓	1	5.c	odd	4	1
90.8.a.i		1	15.e	even	4	1
150.8.a.q		1	5.c	odd	4	1
150.8.c.j		2	1.a	even	1	1	trivial
150.8.c.j		2	5.b	even	2	1	inner
240.8.a.k		1	20.e	even	4	1
450.8.a.k		1	15.e	even	4	1
450.8.c.b		2	3.b	odd	2	1
450.8.c.b		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{2} + 1175056 $ T7^2 + 1175056 acting on $S_{8}^{\mathrm{new}}(150, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 64 $ T^2 + 64
$3$	$ T^{2} + 729 $ T^2 + 729
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 1175056 $ T^2 + 1175056
$11$	$ (T - 6552)^{2} $ (T - 6552)^2
$13$	$ T^{2} + 6360484 $ T^2 + 6360484
$17$	$ T^{2} + 929396196 $ T^2 + 929396196
$19$	$ (T + 12020)^{2} $ (T + 12020)^2
$23$	$ T^{2} + 12446784 $ T^2 + 12446784
$29$	$ (T + 84930)^{2} $ (T + 84930)^2
$31$	$ (T + 94168)^{2} $ (T + 94168)^2
$37$	$ T^{2} + 260073480676 $ T^2 + 260073480676
$41$	$ (T - 841002)^{2} $ (T - 841002)^2
$43$	$ T^{2} + 790413458704 $ T^2 + 790413458704
$47$	$ T^{2} + 726353925696 $ T^2 + 726353925696
$53$	$ T^{2} + 2542372848324 $ T^2 + 2542372848324
$59$	$ (T + 752040)^{2} $ (T + 752040)^2
$61$	$ (T - 1538702)^{2} $ (T - 1538702)^2
$67$	$ T^{2} + 897801730576 $ T^2 + 897801730576
$71$	$ (T + 3824928)^{2} $ (T + 3824928)^2
$73$	$ T^{2} + 834807487684 $ T^2 + 834807487684
$79$	$ (T + 1621880)^{2} $ (T + 1621880)^2
$83$	$ T^{2} + 2160935280144 $ T^2 + 2160935280144
$89$	$ (T + 2375010)^{2} $ (T + 2375010)^2
$97$	$ T^{2} + 199905269553796 $ T^2 + 199905269553796
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 8 i q^{2} - 27 i q^{3} - 64 q^{4} + 216 q^{6} + 1084 i q^{7} - 512 i q^{8} - 729 q^{9} + 6552 q^{11} + 1728 i q^{12} + 2522 i q^{13} - 8672 q^{14} + 4096 q^{16} - 30486 i q^{17} - 5832 i q^{18} - 12020 q^{19} + \cdots - 4776408 q^{99} +O(q^{100}) \) q + 8i q^2 - 27i q^3 - 64 * q^4 + 216 * q^6 + 1084i q^7 - 512i q^8 - 729 * q^9 + 6552 * q^11 + 1728i q^12 + 2522i q^13 - 8672 * q^14 + 4096 * q^16 - 30486i q^17 - 5832i q^18 - 12020 * q^19 + 29268 * q^21 + 52416i q^22 - 3528i q^23 - 13824 * q^24 - 20176 * q^26 + 19683i q^27 - 69376i q^28 - 84930 * q^29 - 94168 * q^31 + 32768i q^32 - 176904i q^33 + 243888 * q^34 + 46656 * q^36 + 509974i q^37 - 96160i q^38 + 68094 * q^39 + 841002 * q^41 + 234144i q^42 + 889052i q^43 - 419328 * q^44 + 28224 * q^46 + 852264i q^47 - 110592i q^48 - 351513 * q^49 - 823122 * q^51 - 161408i q^52 + 1594482i q^53 - 157464 * q^54 + 555008 * q^56 + 324540i q^57 - 679440i q^58 - 752040 * q^59 + 1538702 * q^61 - 753344i q^62 - 790236i q^63 - 262144 * q^64 + 1415232 * q^66 + 947524i q^67 + 1951104i q^68 - 95256 * q^69 - 3824928 * q^71 + 373248i q^72 - 913678i q^73 - 4079792 * q^74 + 769280 * q^76 + 7102368i q^77 + 544752i q^78 - 1621880 * q^79 + 531441 * q^81 + 6728016i q^82 + 1470012i q^83 - 1873152 * q^84 - 7112416 * q^86 + 2293110i q^87 - 3354624i q^88 - 2375010 * q^89 - 2733848 * q^91 + 225792i q^92 + 2542536i q^93 - 6818112 * q^94 + 884736 * q^96 - 14138786i q^97 - 2812104i q^98 - 4776408 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} + 432 q^{6} - 1458 q^{9} + 13104 q^{11} - 17344 q^{14} + 8192 q^{16} - 24040 q^{19} + 58536 q^{21} - 27648 q^{24} - 40352 q^{26} - 169860 q^{29} - 188336 q^{31} + 487776 q^{34} + 93312 q^{36}+ \cdots - 9552816 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 + 432 * q^6 - 1458 * q^9 + 13104 * q^11 - 17344 * q^14 + 8192 * q^16 - 24040 * q^19 + 58536 * q^21 - 27648 * q^24 - 40352 * q^26 - 169860 * q^29 - 188336 * q^31 + 487776 * q^34 + 93312 * q^36 + 136188 * q^39 + 1682004 * q^41 - 838656 * q^44 + 56448 * q^46 - 703026 * q^49 - 1646244 * q^51 - 314928 * q^54 + 1110016 * q^56 - 1504080 * q^59 + 3077404 * q^61 - 524288 * q^64 + 2830464 * q^66 - 190512 * q^69 - 7649856 * q^71 - 8159584 * q^74 + 1538560 * q^76 - 3243760 * q^79 + 1062882 * q^81 - 3746304 * q^84 - 14224832 * q^86 - 4750020 * q^89 - 5467696 * q^91 - 13636224 * q^94 + 1769472 * q^96 - 9552816 * q^99

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