# Properties

 Label 3300.2.c.i Level $3300$ Weight $2$ Character orbit 3300.c Analytic conductor $26.351$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3300.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.3506326670$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 660) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + 2 i q^{7} - q^{9} + q^{11} -2 i q^{13} -2 q^{19} + 2 q^{21} + i q^{27} + 8 q^{31} -i q^{33} + 2 i q^{37} -2 q^{39} -2 i q^{43} + 3 q^{49} -6 i q^{53} + 2 i q^{57} + 12 q^{59} + 2 q^{61} -2 i q^{63} -4 i q^{67} -2 i q^{73} + 2 i q^{77} + 10 q^{79} + q^{81} + 12 i q^{83} + 6 q^{89} + 4 q^{91} -8 i q^{93} + 14 i q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{11} - 4q^{19} + 4q^{21} + 16q^{31} - 4q^{39} + 6q^{49} + 24q^{59} + 4q^{61} + 20q^{79} + 2q^{81} + 12q^{89} + 8q^{91} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1651$$ $$2201$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 2.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 2.00000i 0 −1.00000 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
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 3300.2.c.i 2
3.b odd 2 1 9900.2.c.d 2
5.b even 2 1 inner 3300.2.c.i 2
5.c odd 4 1 660.2.a.d 1
5.c odd 4 1 3300.2.a.b 1
15.d odd 2 1 9900.2.c.d 2
15.e even 4 1 1980.2.a.f 1
15.e even 4 1 9900.2.a.e 1
20.e even 4 1 2640.2.a.b 1
55.e even 4 1 7260.2.a.l 1
60.l odd 4 1 7920.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.d 1 5.c odd 4 1
1980.2.a.f 1 15.e even 4 1
2640.2.a.b 1 20.e even 4 1
3300.2.a.b 1 5.c odd 4 1
3300.2.c.i 2 1.a even 1 1 trivial
3300.2.c.i 2 5.b even 2 1 inner
7260.2.a.l 1 55.e even 4 1
7920.2.a.y 1 60.l odd 4 1
9900.2.a.e 1 15.e even 4 1
9900.2.c.d 2 3.b odd 2 1
9900.2.c.d 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3300, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{13}^{2} + 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$196 + T^{2}$$