# Properties

 Label 3312.2.a.n Level $3312$ Weight $2$ Character orbit 3312.a Self dual yes Analytic conductor $26.446$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3312 = 2^{4} \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3312.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.4464531494$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{5} + 2 q^{7} + O(q^{10})$$ $$q + 2 q^{5} + 2 q^{7} - 6 q^{11} - 2 q^{13} - q^{23} - q^{25} - 6 q^{29} - 8 q^{31} + 4 q^{35} - 10 q^{41} + 12 q^{43} - 8 q^{47} - 3 q^{49} - 2 q^{53} - 12 q^{55} - 12 q^{59} + 4 q^{61} - 4 q^{65} + 12 q^{67} - 10 q^{73} - 12 q^{77} + 6 q^{79} + 14 q^{83} - 4 q^{91} - 6 q^{97} + O(q^{100})$$

## 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 0 2.00000 0 2.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3312.2.a.n 1
3.b odd 2 1 1104.2.a.e 1
4.b odd 2 1 414.2.a.d 1
12.b even 2 1 138.2.a.a 1
24.f even 2 1 4416.2.a.z 1
24.h odd 2 1 4416.2.a.m 1
60.h even 2 1 3450.2.a.y 1
60.l odd 4 2 3450.2.d.j 2
84.h odd 2 1 6762.2.a.q 1
92.b even 2 1 9522.2.a.i 1
276.h odd 2 1 3174.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.a 1 12.b even 2 1
414.2.a.d 1 4.b odd 2 1
1104.2.a.e 1 3.b odd 2 1
3174.2.a.b 1 276.h odd 2 1
3312.2.a.n 1 1.a even 1 1 trivial
3450.2.a.y 1 60.h even 2 1
3450.2.d.j 2 60.l odd 4 2
4416.2.a.m 1 24.h odd 2 1
4416.2.a.z 1 24.f even 2 1
6762.2.a.q 1 84.h odd 2 1
9522.2.a.i 1 92.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3312))$$:

 $$T_{5} - 2$$ $$T_{7} - 2$$ $$T_{11} + 6$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-2 + T$$
$7$ $$-2 + T$$
$11$ $$6 + T$$
$13$ $$2 + T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$1 + T$$
$29$ $$6 + T$$
$31$ $$8 + T$$
$37$ $$T$$
$41$ $$10 + T$$
$43$ $$-12 + T$$
$47$ $$8 + T$$
$53$ $$2 + T$$
$59$ $$12 + T$$
$61$ $$-4 + T$$
$67$ $$-12 + T$$
$71$ $$T$$
$73$ $$10 + T$$
$79$ $$-6 + T$$
$83$ $$-14 + T$$
$89$ $$T$$
$97$ $$6 + T$$