LMFDB - Newform orbit 150.8.c.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,Mod(49,150)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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:	yes
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.

$f(q)$	$=$	$q + 8 i q^{2} + 27 i q^{3} - 64 q^{4} - 216 q^{6} + 1427 i q^{7} - 512 i q^{8} - 729 q^{9} + 5070 q^{11} - 1728 i q^{12} + 11899 i q^{13} - 11416 q^{14} + 4096 q^{16} + 19242 i q^{17} - 5832 i q^{18} + 43711 q^{19} + \cdots - 3696030 q^{99} +O(q^{100})$ q + 8i q^2 + 27i q^3 - 64 * q^4 - 216 * q^6 + 1427i q^7 - 512i q^8 - 729 * q^9 + 5070 * q^11 - 1728i q^12 + 11899i q^13 - 11416 * q^14 + 4096 * q^16 + 19242i q^17 - 5832i q^18 + 43711 * q^19 - 38529 * q^21 + 40560i q^22 + 3150i q^23 + 13824 * q^24 - 95192 * q^26 - 19683i q^27 - 91328i q^28 - 140106 * q^29 + 147563 * q^31 + 32768i q^32 + 136890i q^33 - 153936 * q^34 + 46656 * q^36 + 561674i q^37 + 349688i q^38 - 321273 * q^39 - 270336 * q^41 - 308232i q^42 - 180683i q^43 - 324480 * q^44 - 25200 * q^46 + 97470i q^47 + 110592i q^48 - 1212786 * q^49 - 519534 * q^51 - 761536i q^52 - 2130132i q^53 + 157464 * q^54 + 730624 * q^56 + 1180197i q^57 - 1120848i q^58 + 935070 * q^59 + 135875 * q^61 + 1180504i q^62 - 1040283i q^63 - 262144 * q^64 - 1095120 * q^66 - 1443433i q^67 - 1231488i q^68 - 85050 * q^69 - 2685840 * q^71 + 373248i q^72 - 3280466i q^73 - 4493392 * q^74 - 2797504 * q^76 + 7234890i q^77 - 2570184i q^78 - 5672672 * q^79 + 531441 * q^81 - 2162688i q^82 - 4227906i q^83 + 2465856 * q^84 + 1445464 * q^86 - 3782862i q^87 - 2595840i q^88 + 11890848 * q^89 - 16979873 * q^91 - 201600i q^92 + 3984201i q^93 - 779760 * q^94 - 884736 * q^96 + 3808529i q^97 - 9702288i q^98 - 3696030 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 10140 q^{11} - 22832 q^{14} + 8192 q^{16} + 87422 q^{19} - 77058 q^{21} + 27648 q^{24} - 190384 q^{26} - 280212 q^{29} + 295126 q^{31} - 307872 q^{34} + 93312 q^{36}+ \cdots - 7392060 q^{99}+O(q^{100})$ 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 10140 * q^11 - 22832 * q^14 + 8192 * q^16 + 87422 * q^19 - 77058 * q^21 + 27648 * q^24 - 190384 * q^26 - 280212 * q^29 + 295126 * q^31 - 307872 * q^34 + 93312 * q^36 - 642546 * q^39 - 540672 * q^41 - 648960 * q^44 - 50400 * q^46 - 2425572 * q^49 - 1039068 * q^51 + 314928 * q^54 + 1461248 * q^56 + 1870140 * q^59 + 271750 * q^61 - 524288 * q^64 - 2190240 * q^66 - 170100 * q^69 - 5371680 * q^71 - 8986784 * q^74 - 5595008 * q^76 - 11345344 * q^79 + 1062882 * q^81 + 4931712 * q^84 + 2890928 * q^86 + 23781696 * q^89 - 33959746 * q^91 - 1559520 * q^94 - 1769472 * q^96 - 7392060 * 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

−

1427.00i

512.000i

−729.000

49.2

8.00000i

27.0000i

−64.0000

−216.000

1427.00i

−

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.d		2
3.b	odd	2	1		450.8.c.d		2
5.b	even	2	1	inner	150.8.c.d		2
5.c	odd	4	1		150.8.a.f	✓	1
5.c	odd	4	1		150.8.a.l	yes	1
15.d	odd	2	1		450.8.c.d		2
15.e	even	4	1		450.8.a.m		1
15.e	even	4	1		450.8.a.o		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
150.8.a.f	✓	1	5.c	odd	4	1
150.8.a.l	yes	1	5.c	odd	4	1
150.8.c.d		2	1.a	even	1	1	trivial
150.8.c.d		2	5.b	even	2	1	inner
450.8.a.m		1	15.e	even	4	1
450.8.a.o		1	15.e	even	4	1
450.8.c.d		2	3.b	odd	2	1
450.8.c.d		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 2036329$ T7^2 + 2036329 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
$7$	$T^{2} + 2036329$ T^2 + 2036329
$11$	$(T - 5070)^{2}$ (T - 5070)^2
$13$	$T^{2} + 141586201$ T^2 + 141586201
$17$	$T^{2} + 370254564$ T^2 + 370254564
$19$	$(T - 43711)^{2}$ (T - 43711)^2
$23$	$T^{2} + 9922500$ T^2 + 9922500
$29$	$(T + 140106)^{2}$ (T + 140106)^2
$31$	$(T - 147563)^{2}$ (T - 147563)^2
$37$	$T^{2} + 315477682276$ T^2 + 315477682276
$41$	$(T + 270336)^{2}$ (T + 270336)^2
$43$	$T^{2} + 32646346489$ T^2 + 32646346489
$47$	$T^{2} + 9500400900$ T^2 + 9500400900
$53$	$T^{2} + 4537462337424$ T^2 + 4537462337424
$59$	$(T - 935070)^{2}$ (T - 935070)^2
$61$	$(T - 135875)^{2}$ (T - 135875)^2
$67$	$T^{2} + 2083498825489$ T^2 + 2083498825489
$71$	$(T + 2685840)^{2}$ (T + 2685840)^2
$73$	$T^{2} + 10761457177156$ T^2 + 10761457177156
$79$	$(T + 5672672)^{2}$ (T + 5672672)^2
$83$	$T^{2} + 17875189144836$ T^2 + 17875189144836
$89$	$(T - 11890848)^{2}$ (T - 11890848)^2
$97$	$T^{2} + 14504893143841$ T^2 + 14504893143841
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