LMFDB - Newform orbit 150.8.c.b

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,-120] 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.

$f(q)$	$=$	$q + 8 i q^{2} + 27 i q^{3} - 64 q^{4} - 216 q^{6} + 232 i q^{7} - 512 i q^{8} - 729 q^{9} - 60 q^{11} - 1728 i q^{12} + 14054 i q^{13} - 1856 q^{14} + 4096 q^{16} - 13938 i q^{17} - 5832 i q^{18} - 53564 q^{19} + \cdots + 43740 q^{99} +O(q^{100})$ q + 8i q^2 + 27i q^3 - 64 * q^4 - 216 * q^6 + 232i q^7 - 512i q^8 - 729 * q^9 - 60 * q^11 - 1728i q^12 + 14054i q^13 - 1856 * q^14 + 4096 * q^16 - 13938i q^17 - 5832i q^18 - 53564 * q^19 - 6264 * q^21 - 480i q^22 - 27000i q^23 + 13824 * q^24 - 112432 * q^26 - 19683i q^27 - 14848i q^28 + 88554 * q^29 - 120352 * q^31 + 32768i q^32 - 1620i q^33 + 111504 * q^34 + 46656 * q^36 - 469646i q^37 - 428512i q^38 - 379458 * q^39 - 510246 * q^41 - 50112i q^42 + 145412i q^43 + 3840 * q^44 + 216000 * q^46 - 304560i q^47 + 110592i q^48 + 769719 * q^49 + 376326 * q^51 - 899456i q^52 + 94398i q^53 + 157464 * q^54 + 118784 * q^56 - 1446228i q^57 + 708432i q^58 + 1885740 * q^59 - 271690 * q^61 - 962816i q^62 - 169128i q^63 - 262144 * q^64 + 12960 * q^66 - 1074668i q^67 + 892032i q^68 + 729000 * q^69 - 2505480 * q^71 + 373248i q^72 - 2954566i q^73 + 3757168 * q^74 + 3428096 * q^76 - 13920i q^77 - 3035664i q^78 + 7354768 * q^79 + 531441 * q^81 - 4081968i q^82 - 3846756i q^83 + 400896 * q^84 - 1163296 * q^86 + 2390958i q^87 + 30720i q^88 - 5685162 * q^89 - 3260528 * q^91 + 1728000i q^92 - 3249504i q^93 + 2436480 * q^94 - 884736 * q^96 - 7981346i q^97 + 6157752i q^98 + 43740 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} - 120 q^{11} - 3712 q^{14} + 8192 q^{16} - 107128 q^{19} - 12528 q^{21} + 27648 q^{24} - 224864 q^{26} + 177108 q^{29} - 240704 q^{31} + 223008 q^{34} + 93312 q^{36}+ \cdots + 87480 q^{99}+O(q^{100})$ 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 - 120 * q^11 - 3712 * q^14 + 8192 * q^16 - 107128 * q^19 - 12528 * q^21 + 27648 * q^24 - 224864 * q^26 + 177108 * q^29 - 240704 * q^31 + 223008 * q^34 + 93312 * q^36 - 758916 * q^39 - 1020492 * q^41 + 7680 * q^44 + 432000 * q^46 + 1539438 * q^49 + 752652 * q^51 + 314928 * q^54 + 237568 * q^56 + 3771480 * q^59 - 543380 * q^61 - 524288 * q^64 + 25920 * q^66 + 1458000 * q^69 - 5010960 * q^71 + 7514336 * q^74 + 6856192 * q^76 + 14709536 * q^79 + 1062882 * q^81 + 801792 * q^84 - 2326592 * q^86 - 11370324 * q^89 - 6521056 * q^91 + 4872960 * q^94 - 1769472 * q^96 + 87480 * 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

−

232.000i

512.000i

−729.000

49.2

8.00000i

27.0000i

−64.0000

−216.000

232.000i

−

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 53824$ T7^2 + 53824 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
$7$	$T^{2} + 53824$ T^2 + 53824
$11$	$(T + 60)^{2}$ (T + 60)^2
$13$	$T^{2} + 197514916$ T^2 + 197514916
$17$	$T^{2} + 194267844$ T^2 + 194267844
$19$	$(T + 53564)^{2}$ (T + 53564)^2
$23$	$T^{2} + 729000000$ T^2 + 729000000
$29$	$(T - 88554)^{2}$ (T - 88554)^2
$31$	$(T + 120352)^{2}$ (T + 120352)^2
$37$	$T^{2} + 220567365316$ T^2 + 220567365316
$41$	$(T + 510246)^{2}$ (T + 510246)^2
$43$	$T^{2} + 21144649744$ T^2 + 21144649744
$47$	$T^{2} + 92756793600$ T^2 + 92756793600
$53$	$T^{2} + 8910982404$ T^2 + 8910982404
$59$	$(T - 1885740)^{2}$ (T - 1885740)^2
$61$	$(T + 271690)^{2}$ (T + 271690)^2
$67$	$T^{2} + 1154911310224$ T^2 + 1154911310224
$71$	$(T + 2505480)^{2}$ (T + 2505480)^2
$73$	$T^{2} + 8729460248356$ T^2 + 8729460248356
$79$	$(T - 7354768)^{2}$ (T - 7354768)^2
$83$	$T^{2} + 14797531723536$ T^2 + 14797531723536
$89$	$(T + 5685162)^{2}$ (T + 5685162)^2
$97$	$T^{2} + 63701883971716$ T^2 + 63701883971716
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