Properties

 Label 3267.2.a.f Level $3267$ Weight $2$ Character orbit 3267.a Self dual yes Analytic conductor $26.087$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3267.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$26.0871263404$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + q^{7}+O(q^{10})$$ q - 2 * q^4 + q^7 $$q - 2 q^{4} + q^{7} - 5 q^{13} + 4 q^{16} + 7 q^{19} - 5 q^{25} - 2 q^{28} - 4 q^{31} + 11 q^{37} - 8 q^{43} - 6 q^{49} + 10 q^{52} + q^{61} - 8 q^{64} + 5 q^{67} + 7 q^{73} - 14 q^{76} - 17 q^{79} - 5 q^{91} - 19 q^{97}+O(q^{100})$$ q - 2 * q^4 + q^7 - 5 * q^13 + 4 * q^16 + 7 * q^19 - 5 * q^25 - 2 * q^28 - 4 * q^31 + 11 * q^37 - 8 * q^43 - 6 * q^49 + 10 * q^52 + q^61 - 8 * q^64 + 5 * q^67 + 7 * q^73 - 14 * q^76 - 17 * q^79 - 5 * q^91 - 19 * q^97

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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 −2.00000 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.2.a.f 1
3.b odd 2 1 CM 3267.2.a.f 1
11.b odd 2 1 27.2.a.a 1
33.d even 2 1 27.2.a.a 1
44.c even 2 1 432.2.a.e 1
55.d odd 2 1 675.2.a.e 1
55.e even 4 2 675.2.b.f 2
77.b even 2 1 1323.2.a.i 1
88.b odd 2 1 1728.2.a.n 1
88.g even 2 1 1728.2.a.o 1
99.g even 6 2 81.2.c.a 2
99.h odd 6 2 81.2.c.a 2
132.d odd 2 1 432.2.a.e 1
143.d odd 2 1 4563.2.a.e 1
165.d even 2 1 675.2.a.e 1
165.l odd 4 2 675.2.b.f 2
187.b odd 2 1 7803.2.a.k 1
209.d even 2 1 9747.2.a.f 1
231.h odd 2 1 1323.2.a.i 1
264.m even 2 1 1728.2.a.n 1
264.p odd 2 1 1728.2.a.o 1
297.o even 18 6 729.2.e.f 6
297.q odd 18 6 729.2.e.f 6
396.k even 6 2 1296.2.i.i 2
396.o odd 6 2 1296.2.i.i 2
429.e even 2 1 4563.2.a.e 1
561.h even 2 1 7803.2.a.k 1
627.b odd 2 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 11.b odd 2 1
27.2.a.a 1 33.d even 2 1
81.2.c.a 2 99.g even 6 2
81.2.c.a 2 99.h odd 6 2
432.2.a.e 1 44.c even 2 1
432.2.a.e 1 132.d odd 2 1
675.2.a.e 1 55.d odd 2 1
675.2.a.e 1 165.d even 2 1
675.2.b.f 2 55.e even 4 2
675.2.b.f 2 165.l odd 4 2
729.2.e.f 6 297.o even 18 6
729.2.e.f 6 297.q odd 18 6
1296.2.i.i 2 396.k even 6 2
1296.2.i.i 2 396.o odd 6 2
1323.2.a.i 1 77.b even 2 1
1323.2.a.i 1 231.h odd 2 1
1728.2.a.n 1 88.b odd 2 1
1728.2.a.n 1 264.m even 2 1
1728.2.a.o 1 88.g even 2 1
1728.2.a.o 1 264.p odd 2 1
3267.2.a.f 1 1.a even 1 1 trivial
3267.2.a.f 1 3.b odd 2 1 CM
4563.2.a.e 1 143.d odd 2 1
4563.2.a.e 1 429.e even 2 1
7803.2.a.k 1 187.b odd 2 1
7803.2.a.k 1 561.h even 2 1
9747.2.a.f 1 209.d even 2 1
9747.2.a.f 1 627.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3267))$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7} - 1$$ T7 - 1 $$T_{23}$$ T23

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 5$$
$17$ $$T$$
$19$ $$T - 7$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T - 11$$
$41$ $$T$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 1$$
$67$ $$T - 5$$
$71$ $$T$$
$73$ $$T - 7$$
$79$ $$T + 17$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 19$$