# Properties

 Label 16.9.c.b Level $16$ Weight $9$ Character orbit 16.c Analytic conductor $6.518$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,9,Mod(15,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("16.15");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$6.51805776098$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 $$\beta = 8\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 258 q^{5} - 238 \beta q^{7} + 6369 q^{9} +O(q^{10})$$ q - b * q^3 + 258 * q^5 - 238*b * q^7 + 6369 * q^9 $$q - \beta q^{3} + 258 q^{5} - 238 \beta q^{7} + 6369 q^{9} - 1671 \beta q^{11} + 19138 q^{13} - 258 \beta q^{15} - 58686 q^{17} + 11011 \beta q^{19} - 45696 q^{21} + 19494 \beta q^{23} - 324061 q^{25} - 12930 \beta q^{27} + 842178 q^{29} - 75864 \beta q^{31} - 320832 q^{33} - 61404 \beta q^{35} + 2548610 q^{37} - 19138 \beta q^{39} - 4324158 q^{41} + 147009 \beta q^{43} + 1643202 q^{45} + 522012 \beta q^{47} - 5110847 q^{49} + 58686 \beta q^{51} + 1192194 q^{53} - 431118 \beta q^{55} + 2114112 q^{57} - 24411 \beta q^{59} + 8414786 q^{61} - 1515822 \beta q^{63} + 4937604 q^{65} - 1257521 \beta q^{67} + 3742848 q^{69} + 2228322 \beta q^{71} + 12735874 q^{73} + 324061 \beta q^{75} - 76358016 q^{77} + 453700 \beta q^{79} + 39304449 q^{81} + 5995347 \beta q^{83} - 15140988 q^{85} - 842178 \beta q^{87} - 16802814 q^{89} - 4554844 \beta q^{91} - 14565888 q^{93} + 2840838 \beta q^{95} + 120994882 q^{97} - 10642599 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 258 * q^5 - 238*b * q^7 + 6369 * q^9 - 1671*b * q^11 + 19138 * q^13 - 258*b * q^15 - 58686 * q^17 + 11011*b * q^19 - 45696 * q^21 + 19494*b * q^23 - 324061 * q^25 - 12930*b * q^27 + 842178 * q^29 - 75864*b * q^31 - 320832 * q^33 - 61404*b * q^35 + 2548610 * q^37 - 19138*b * q^39 - 4324158 * q^41 + 147009*b * q^43 + 1643202 * q^45 + 522012*b * q^47 - 5110847 * q^49 + 58686*b * q^51 + 1192194 * q^53 - 431118*b * q^55 + 2114112 * q^57 - 24411*b * q^59 + 8414786 * q^61 - 1515822*b * q^63 + 4937604 * q^65 - 1257521*b * q^67 + 3742848 * q^69 + 2228322*b * q^71 + 12735874 * q^73 + 324061*b * q^75 - 76358016 * q^77 + 453700*b * q^79 + 39304449 * q^81 + 5995347*b * q^83 - 15140988 * q^85 - 842178*b * q^87 - 16802814 * q^89 - 4554844*b * q^91 - 14565888 * q^93 + 2840838*b * q^95 + 120994882 * q^97 - 10642599*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 516 q^{5} + 12738 q^{9}+O(q^{10})$$ 2 * q + 516 * q^5 + 12738 * q^9 $$2 q + 516 q^{5} + 12738 q^{9} + 38276 q^{13} - 117372 q^{17} - 91392 q^{21} - 648122 q^{25} + 1684356 q^{29} - 641664 q^{33} + 5097220 q^{37} - 8648316 q^{41} + 3286404 q^{45} - 10221694 q^{49} + 2384388 q^{53} + 4228224 q^{57} + 16829572 q^{61} + 9875208 q^{65} + 7485696 q^{69} + 25471748 q^{73} - 152716032 q^{77} + 78608898 q^{81} - 30281976 q^{85} - 33605628 q^{89} - 29131776 q^{93} + 241989764 q^{97}+O(q^{100})$$ 2 * q + 516 * q^5 + 12738 * q^9 + 38276 * q^13 - 117372 * q^17 - 91392 * q^21 - 648122 * q^25 + 1684356 * q^29 - 641664 * q^33 + 5097220 * q^37 - 8648316 * q^41 + 3286404 * q^45 - 10221694 * q^49 + 2384388 * q^53 + 4228224 * q^57 + 16829572 * q^61 + 9875208 * q^65 + 7485696 * q^69 + 25471748 * q^73 - 152716032 * q^77 + 78608898 * q^81 - 30281976 * q^85 - 33605628 * q^89 - 29131776 * q^93 + 241989764 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 13.8564i 0 258.000 0 3297.82i 0 6369.00 0
15.2 0 13.8564i 0 258.000 0 3297.82i 0 6369.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.9.c.b 2
3.b odd 2 1 144.9.g.d 2
4.b odd 2 1 inner 16.9.c.b 2
5.b even 2 1 400.9.b.e 2
5.c odd 4 2 400.9.h.a 4
8.b even 2 1 64.9.c.c 2
8.d odd 2 1 64.9.c.c 2
12.b even 2 1 144.9.g.d 2
16.e even 4 2 256.9.d.d 4
16.f odd 4 2 256.9.d.d 4
20.d odd 2 1 400.9.b.e 2
20.e even 4 2 400.9.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.b 2 1.a even 1 1 trivial
16.9.c.b 2 4.b odd 2 1 inner
64.9.c.c 2 8.b even 2 1
64.9.c.c 2 8.d odd 2 1
144.9.g.d 2 3.b odd 2 1
144.9.g.d 2 12.b even 2 1
256.9.d.d 4 16.e even 4 2
256.9.d.d 4 16.f odd 4 2
400.9.b.e 2 5.b even 2 1
400.9.b.e 2 20.d odd 2 1
400.9.h.a 4 5.c odd 4 2
400.9.h.a 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 192$$ acting on $$S_{9}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 192$$
$5$ $$(T - 258)^{2}$$
$7$ $$T^{2} + 10875648$$
$11$ $$T^{2} + 536110272$$
$13$ $$(T - 19138)^{2}$$
$17$ $$(T + 58686)^{2}$$
$19$ $$T^{2} + 23278487232$$
$23$ $$T^{2} + 72963078912$$
$29$ $$(T - 842178)^{2}$$
$31$ $$T^{2} + 1105026527232$$
$37$ $$(T - 2548610)^{2}$$
$41$ $$(T + 4324158)^{2}$$
$43$ $$T^{2} + 4149436047552$$
$47$ $$T^{2} + 52319333403648$$
$53$ $$(T - 1192194)^{2}$$
$59$ $$T^{2} + 114412208832$$
$61$ $$(T - 8414786)^{2}$$
$67$ $$T^{2} + \cdots + 303620940564672$$
$71$ $$T^{2} + \cdots + 953360435651328$$
$73$ $$(T - 12735874)^{2}$$
$79$ $$T^{2} + 39521988480000$$
$83$ $$T^{2} + 69\!\cdots\!28$$
$89$ $$(T + 16802814)^{2}$$
$97$ $$(T - 120994882)^{2}$$