LMFDB - Newform orbit 90.16.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,16,Mod(19,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("90.19");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$N$	$=$	$90 = 2 \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$16$
Character orbit:	$[\chi]$	$=$	90.c (of order $2$ , degree $1$ , 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:	$128.424154590$
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:	$\Z[a_1, \ldots, a_{5}]$
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.

$f(q)$	$=$	$q - 128 i q^{2} - 16384 q^{4} + ( - 171875 i + 31250) q^{5} + 511994 i q^{7} + 2097152 i q^{8} + ( - 4000000 i - 22000000) q^{10} + 19947598 q^{11} + 180100962 i q^{13} + 65535232 q^{14} + 268435456 q^{16} + \cdots - 574134227700096 i q^{98} +O(q^{100})$ q - 128i q^2 - 16384 * q^4 + (-171875i + 31250) q^5 + 511994i q^7 + 2097152i q^8 + (-4000000i - 22000000) q^10 + 19947598 * q^11 + 180100962i q^13 + 65535232 * q^14 + 268435456 * q^16 - 565654514i q^17 + 2169230520 * q^19 + (2816000000i - 512000000) q^20 - 2553292544i q^22 - 3443072572i q^23 + (-10742187500i - 28564453125) q^25 + 23052923136 * q^26 - 8388509696i q^28 + 58843361520 * q^29 - 131248934648 * q^31 - 34359738368i q^32 - 72403777792 * q^34 + (15999812500i + 87998968750) q^35 - 1018926148246i q^37 - 277661506560i q^38 + (65536000000i + 360448000000) q^40 + 678311212798 * q^41 - 1869778918508i q^43 - 326821445632 * q^44 - 440713289216 * q^46 + 2655546152576i q^47 + 4485423653907 * q^49 + (3656250000000i - 1375000000000) q^50 - 2950774161408i q^52 + 7733409097998i q^53 + (-3428493406250i + 623362437500) q^55 - 1073729241088 * q^56 - 7531950274560i q^58 + 12384358030090 * q^59 - 28742040118198 * q^61 + 16799863634944i q^62 - 4398046511104 * q^64 + (5628155062500i + 30954852843750) q^65 - 62114558153336i q^67 + 9267683557376i q^68 + (-11263868000000i + 2047976000000) q^70 - 17045715067452 * q^71 - 94423385896028i q^73 - 130422546975488 * q^74 - 35540672839680 * q^76 + 10213050490412i q^77 - 5162001412320 * q^79 + (-46137344000000i + 8388608000000) q^80 - 86823835238144i q^82 - 388824931818532i q^83 + (-17676703562500i - 97221869593750) q^85 - 239331701569024 * q^86 + 41833145040896i q^88 + 64970078898710 * q^89 - 92210611938228 * q^91 + 56411301019648i q^92 + 339909907529728 * q^94 + (-372836495625000i + 67788453750000) q^95 - 424522131387176i q^97 - 574134227700096i q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 32768 q^{4} + 62500 q^{5} - 44000000 q^{10} + 39895196 q^{11} + 131070464 q^{14} + 536870912 q^{16} + 4338461040 q^{19} - 1024000000 q^{20} - 57128906250 q^{25} + 46105846272 q^{26} + 117686723040 q^{29}+ \cdots + 135576907500000 q^{95}+O(q^{100})$ 2 * q - 32768 * q^4 + 62500 * q^5 - 44000000 * q^10 + 39895196 * q^11 + 131070464 * q^14 + 536870912 * q^16 + 4338461040 * q^19 - 1024000000 * q^20 - 57128906250 * q^25 + 46105846272 * q^26 + 117686723040 * q^29 - 262497869296 * q^31 - 144807555584 * q^34 + 175997937500 * q^35 + 720896000000 * q^40 + 1356622425596 * q^41 - 653642891264 * q^44 - 881426578432 * q^46 + 8970847307814 * q^49 - 2750000000000 * q^50 + 1246724875000 * q^55 - 2147458482176 * q^56 + 24768716060180 * q^59 - 57484080236396 * q^61 - 8796093022208 * q^64 + 61909705687500 * q^65 + 4095952000000 * q^70 - 34091430134904 * q^71 - 260845093950976 * q^74 - 71081345679360 * q^76 - 10324002824640 * q^79 + 16777216000000 * q^80 - 194443739187500 * q^85 - 478663403138048 * q^86 + 129940157797420 * q^89 - 184421223876456 * q^91 + 679819815059456 * q^94 + 135576907500000 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$ .

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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}

19.1

		1.00000i
	−	1.00000i

−

128.000i

−16384.0

31250.0

−

171875.i

511994.i

2.09715e6i

−2.20000e7

−

4.00000e6i

19.2

128.000i

−16384.0

31250.0

171875.i

−

511994.i

−

2.09715e6i

−2.20000e7

4.00000e6i

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	90.16.c.a		2
3.b	odd	2	1		30.16.c.a	✓	2
5.b	even	2	1	inner	90.16.c.a		2
15.d	odd	2	1		30.16.c.a	✓	2
15.e	even	4	1		150.16.a.b		1
15.e	even	4	1		150.16.a.i		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.16.c.a	✓	2	3.b	odd	2	1
30.16.c.a	✓	2	15.d	odd	2	1
90.16.c.a		2	1.a	even	1	1	trivial
90.16.c.a		2	5.b	even	2	1	inner
150.16.a.b		1	15.e	even	4	1
150.16.a.i		1	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 262137856036$ T7^2 + 262137856036 acting on $S_{16}^{\mathrm{new}}(90, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 16384$ T^2 + 16384
$3$	$T^{2}$ T^2
$5$	$T^{2} + \cdots + 30517578125$ T^2 - 62500*T + 30517578125
$7$	$T^{2} + 262137856036$ T^2 + 262137856036
$11$	$(T - 19947598)^{2}$ (T - 19947598)^2
$13$	$T^{2} + 32\!\cdots\!44$ T^2 + 32436356513325444
$17$	$T^{2} + 31\!\cdots\!96$ T^2 + 319965029208576196
$19$	$(T - 2169230520)^{2}$ (T - 2169230520)^2
$23$	$T^{2} + 11\!\cdots\!84$ T^2 + 11854748736058695184
$29$	$(T - 58843361520)^{2}$ (T - 58843361520)^2
$31$	$(T + 131248934648)^{2}$ (T + 131248934648)^2
$37$	$T^{2} + 10\!\cdots\!16$ T^2 + 1038210495579429568876516
$41$	$(T - 678311212798)^{2}$ (T - 678311212798)^2
$43$	$T^{2} + 34\!\cdots\!64$ T^2 + 3496073204096946104946064
$47$	$T^{2} + 70\!\cdots\!76$ T^2 + 7051925368461196271435776
$53$	$T^{2} + 59\!\cdots\!04$ T^2 + 59805616276998239967608004
$59$	$(T - 12384358030090)^{2}$ (T - 12384358030090)^2
$61$	$(T + 28742040118198)^{2}$ (T + 28742040118198)^2
$67$	$T^{2} + 38\!\cdots\!96$ T^2 + 3858218334584159754487928896
$71$	$(T + 17045715067452)^{2}$ (T + 17045715067452)^2
$73$	$T^{2} + 89\!\cdots\!84$ T^2 + 8915775804070219432426176784
$79$	$(T + 5162001412320)^{2}$ (T + 5162001412320)^2
$83$	$T^{2} + 15\!\cdots\!24$ T^2 + 151184827603686058512578635024
$89$	$(T - 64970078898710)^{2}$ (T - 64970078898710)^2
$97$	$T^{2} + 18\!\cdots\!76$ T^2 + 180219040037510722334017254976
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